![Page 1: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/1.jpg)
What is a group?
Scott Harper
Postgraduate Seminar
2nd November 2018
![Page 2: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/2.jpg)
What is a group?
It depends who you ask.
Groups are viewed from many different perspectives by a diversity ofpeople with a wide range of differing interests. Sometimes two seeminglyunrelated properties of a group, coming from two radically distinctviewpoints, miraculously happen to be equivalent. I’ll tell you about someof the different perspectives on group theory and then focus on one casewhere a surprising bridge was built, the Muller–Schupp theorem.
I expect the audience to arrive knowing what a group is and leave doubtingthat they ever did.
![Page 3: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/3.jpg)
What is a group?
It depends who you ask.
Groups are viewed from many different perspectives by a diversity ofpeople with a wide range of differing interests. Sometimes two seeminglyunrelated properties of a group, coming from two radically distinctviewpoints, miraculously happen to be equivalent. I’ll tell you about someof the different perspectives on group theory and then focus on one casewhere a surprising bridge was built, the Muller–Schupp theorem.
I expect the audience to arrive knowing what a group is and leave doubtingthat they ever did.
![Page 4: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/4.jpg)
![Page 5: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/5.jpg)
![Page 6: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/6.jpg)
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
![Page 7: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/7.jpg)
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
![Page 8: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/8.jpg)
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
![Page 9: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/9.jpg)
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
![Page 10: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/10.jpg)
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
a
b
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
![Page 11: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/11.jpg)
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
a
b
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
![Page 12: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/12.jpg)
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
![Page 13: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/13.jpg)
Abstract groups
A group is a set together with a binary operation satisfying . . .
D8 = 1, a, a2, a3, b, ab, a2b, a3b
× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
![Page 14: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/14.jpg)
Abstract groups
A group is a set together with a binary operation satisfying . . .
D8 = 1, a, a2, a3, b, ab, a2b, a3b
× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
![Page 15: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/15.jpg)
Abstract groups
A group is a set together with a binary operation satisfying . . .
D8 = 1, a, a2, a3, b, ab, a2b, a3b
× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
![Page 16: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/16.jpg)
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
![Page 17: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/17.jpg)
Permutation groups
1
23
4
ab
symmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
![Page 18: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/18.jpg)
Permutation groups
1
23
4
ab
symmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
![Page 19: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/19.jpg)
Permutation groups
1
23
4
ab
symmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
![Page 20: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/20.jpg)
Permutation groups
1
23
4
ab
symmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
![Page 21: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/21.jpg)
Permutation groups
1
23
4
ab
symmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
![Page 22: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/22.jpg)
Permutation groups
1
23
4
absymmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
![Page 23: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/23.jpg)
Permutation groups
1
23
4
absymmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
![Page 24: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/24.jpg)
Permutation groups
1
23
4
absymmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
![Page 25: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/25.jpg)
Permutation groups
1
23
4
absymmetries of an oblong
1
23
41
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
![Page 26: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/26.jpg)
Permutation groups
1
23
4
absymmetries of an oblong
1
23
41
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
![Page 27: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/27.jpg)
![Page 28: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/28.jpg)
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
![Page 29: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/29.jpg)
Matrix groups
ab
a =
(0 −11 0
)b =
(−1 00 1
)
D8 =
⟨(0 −11 0
),
(−1 00 1
)⟩6 GL2(R)
A representation is a homomorphism ϕ : G→ GLd(F).
![Page 30: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/30.jpg)
Matrix groups
ab
a =
(0 −11 0
)b =
(−1 00 1
)
D8 =
⟨(0 −11 0
),
(−1 00 1
)⟩6 GL2(R)
A representation is a homomorphism ϕ : G→ GLd(F).
![Page 31: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/31.jpg)
Matrix groups
ab
a =
(0 −11 0
)b =
(−1 00 1
)
D8 =
⟨(0 −11 0
),
(−1 00 1
)⟩6 GL2(R)
A representation is a homomorphism ϕ : G→ GLd(F).
![Page 32: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/32.jpg)
Matrix groups
ab
a =
(0 −11 0
)b =
(−1 00 1
)
D8 =
⟨(0 −11 0
),
(−1 00 1
)⟩6 GL2(R)
A representation is a homomorphism ϕ : G→ GLd(F).
![Page 33: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/33.jpg)
Representation theory
ab
a =
(0 −11 0
)b =
(−1 00 1
)
D8 =
⟨(0 −11 0
),
(−1 00 1
)⟩6 GL2(R)
A representation is a homomorphism ϕ : G→ GLd(F).
![Page 34: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/34.jpg)
![Page 35: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/35.jpg)
Theorem (Burnside, 1904)Let p and q be prime. Any group of order paqb is soluble.
William Burnside
![Page 36: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/36.jpg)
Theorem (Burnside, 1904)Let p and q be prime. Any group of order paqb is soluble.
Ambrose Burnside
![Page 37: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/37.jpg)
![Page 38: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/38.jpg)
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
![Page 39: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/39.jpg)
Geometric group theory
1
aa2
a3
ba3b
a2b ab
Cayley graph of D8 with respect to a, b
Γ(D8, a, b)
![Page 40: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/40.jpg)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
1
aa2
a3
ba3b
a2b ab
Γ(D8, a, b)
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
![Page 41: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/41.jpg)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
1
aa2
a3
ba3b
a2b ab
Γ(D8, a, b)
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
![Page 42: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/42.jpg)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
![Page 43: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/43.jpg)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
bxbx
bx
1
xb
xbxbbxbxb
bxbb
x
xbx
Γ(D∞, a, b)
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
![Page 44: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/44.jpg)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
bxbx
bx
1
xb
xbxbbxbxb
bxbb
x
xbx
Γ(D∞, , b)
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
![Page 45: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/45.jpg)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
bxbx
bx
1
xb
xbxbbxbxb
bxbb
x
xbx
Γ(D∞, , b)
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
![Page 46: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/46.jpg)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
![Page 47: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/47.jpg)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
Different generating sets givedifferent Cayley graphs
– Think up to quasiisometry
Different groups have the sameCayley graph
– Get used to it
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
![Page 48: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/48.jpg)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
Different generating sets givedifferent Cayley graphs
– Think up to quasiisometry
Different groups have the sameCayley graph
– Get used to it
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
![Page 49: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/49.jpg)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
Different generating sets givedifferent Cayley graphs
– Think up to quasiisometry
Different groups have the sameCayley graph
– Get used to it
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
![Page 50: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/50.jpg)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
Different generating sets givedifferent Cayley graphs
– Think up to quasiisometry
Different groups have the sameCayley graph
– Get used to it
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
![Page 51: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/51.jpg)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
Different generating sets givedifferent Cayley graphs
– Think up to quasiisometry
Different groups have the sameCayley graph
– Get used to it
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
![Page 52: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/52.jpg)
“If H is a finite index subgroup of G, then H looks the same as G.”– a geometric group theorist
The index of a subgroup H 6 G is the cardinality of the set Hg | g ∈ G.
〈a〉 is a finite index subgroup of D∞ = 〈a, b〉
If H is a finite index subgroup of G, then Γ(H) is quasiisometric to Γ(G).
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
is quasiisometric to −3 −2 −1 0 1 2 3
A group is virtually purple if it has finite index subgroup that is purple.
〈a〉 is cyclic so D∞ is virtually cyclic
![Page 53: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/53.jpg)
“If H is a finite index subgroup of G, then H looks the same as G.”– a geometric group theorist
The index of a subgroup H 6 G is the cardinality of the set Hg | g ∈ G.
〈a〉 is a finite index subgroup of D∞ = 〈a, b〉
If H is a finite index subgroup of G, then Γ(H) is quasiisometric to Γ(G).
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
is quasiisometric to −3 −2 −1 0 1 2 3
A group is virtually purple if it has finite index subgroup that is purple.
〈a〉 is cyclic so D∞ is virtually cyclic
![Page 54: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/54.jpg)
“If H is a finite index subgroup of G, then H looks the same as G.”– a geometric group theorist
The index of a subgroup H 6 G is the cardinality of the set Hg | g ∈ G.
〈a〉 is a finite index subgroup of D∞ = 〈a, b〉
If H is a finite index subgroup of G, then Γ(H) is quasiisometric to Γ(G).
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
is quasiisometric to −3 −2 −1 0 1 2 3
A group is virtually purple if it has finite index subgroup that is purple.
〈a〉 is cyclic so D∞ is virtually cyclic
![Page 55: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/55.jpg)
“If H is a finite index subgroup of G, then H looks the same as G.”– a geometric group theorist
The index of a subgroup H 6 G is the cardinality of the set Hg | g ∈ G.
〈a〉 is a finite index subgroup of D∞ = 〈a, b〉
If H is a finite index subgroup of G, then Γ(H) is quasiisometric to Γ(G).
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
is quasiisometric to −3 −2 −1 0 1 2 3
A group is virtually purple if it has finite index subgroup that is purple.
〈a〉 is cyclic so D∞ is virtually cyclic
![Page 56: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/56.jpg)
“If H is a finite index subgroup of G, then H looks the same as G.”– a geometric group theorist
The index of a subgroup H 6 G is the cardinality of the set Hg | g ∈ G.
〈a〉 is a finite index subgroup of D∞ = 〈a, b〉
If H is a finite index subgroup of G, then Γ(H) is quasiisometric to Γ(G).
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
is quasiisometric to −3 −2 −1 0 1 2 3
A group is virtually purple if it has finite index subgroup that is purple.
〈a〉 is cyclic so D∞ is virtually cyclic
![Page 57: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/57.jpg)
“If H is a finite index subgroup of G, then H looks the same as G.”– a geometric group theorist
The index of a subgroup H 6 G is the cardinality of the set Hg | g ∈ G.
〈a〉 is a finite index subgroup of D∞ = 〈a, b〉
If H is a finite index subgroup of G, then Γ(H) is quasiisometric to Γ(G).
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
is quasiisometric to −3 −2 −1 0 1 2 3
A group is virtually purple if it has finite index subgroup that is purple.
〈a〉 is cyclic so D∞ is virtually cyclic
![Page 58: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/58.jpg)
“If H is a finite index subgroup of G, then H looks the same as G.”– a geometric group theorist
The index of a subgroup H 6 G is the cardinality of the set Hg | g ∈ G.
〈a〉 is a finite index subgroup of D∞ = 〈a, b〉
If H is a finite index subgroup of G, then Γ(H) is quasiisometric to Γ(G).
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
is quasiisometric to −3 −2 −1 0 1 2 3
A group is virtually purple if it has finite index subgroup that is purple.
〈a〉 is cyclic so D∞ is virtually cyclic
![Page 59: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/59.jpg)
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
![Page 60: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/60.jpg)
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉
= C4 ∼= Z/4Z
〈a | 〉
∼= Z ∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
![Page 61: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/61.jpg)
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉
= C4 ∼= Z/4Z
〈a | 〉
∼= Z ∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
![Page 62: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/62.jpg)
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉
= C4 ∼= Z/4Z
〈a | 〉
∼= Z ∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
![Page 63: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/63.jpg)
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉
= C4 ∼= Z/4Z
〈a | 〉
∼= Z ∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
![Page 64: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/64.jpg)
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉
∼= Z ∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
![Page 65: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/65.jpg)
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉
∼= Z ∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
![Page 66: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/66.jpg)
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉 ∼= Z
∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
![Page 67: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/67.jpg)
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉 ∼= Z
∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
![Page 68: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/68.jpg)
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉 ∼= Z
∼= F1
〈a, b | b2 = 1, ba = a−1b〉 = D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
![Page 69: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/69.jpg)
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉 ∼= Z
∼= F1
〈a, b | b2 = 1, ba = a−1b〉 = D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
![Page 70: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/70.jpg)
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉 ∼= Z
∼= F1
〈a, b | b2 = 1, ba = a−1b〉 = D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
![Page 71: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/71.jpg)
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉 ∼= Z
∼= F1
〈a, b | b2 = 1, ba = a−1b〉 = D∞
〈a, b | 〉 = F2
A group is free if it admits a presentation of the form 〈A | 〉.
![Page 72: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/72.jpg)
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉 ∼= Z ∼= F1
〈a, b | b2 = 1, ba = a−1b〉 = D∞
〈a, b | 〉 = F2
A group is free if it admits a presentation of the form 〈A | 〉.
![Page 73: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/73.jpg)
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
![Page 74: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/74.jpg)
The problem with presentations
Tell me something about this group
〈a, b, c | ab = b2a, bc = c2b, ca = a2c〉
It’s trivial.
The sad realityDetermining whether a presentation gives the trivial group is undecidable.
The Word Problem (Dehn, 1911)Fix a finitely generated group G = 〈A〉. Determine whether a word in A istrivial in G.
Theorem (Novikov, 1954)There are groups for which the Word Problem is undecidable.
![Page 75: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/75.jpg)
The problem with presentations
Tell me something about this group
〈a, b, c | ab = b2a, bc = c2b, ca = a2c〉
It’s trivial.
The sad realityDetermining whether a presentation gives the trivial group is undecidable.
The Word Problem (Dehn, 1911)Fix a finitely generated group G = 〈A〉. Determine whether a word in A istrivial in G.
Theorem (Novikov, 1954)There are groups for which the Word Problem is undecidable.
![Page 76: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/76.jpg)
The problem with presentations
Tell me something about this group
〈a, b, c | ab = b2a, bc = c2b, ca = a2c〉
It’s trivial.
The sad realityDetermining whether a presentation gives the trivial group is undecidable.
The Word Problem (Dehn, 1911)Fix a finitely generated group G = 〈A〉. Determine whether a word in A istrivial in G.
Theorem (Novikov, 1954)There are groups for which the Word Problem is undecidable.
![Page 77: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/77.jpg)
The problem with presentations
Tell me something about this group
〈a, b, c | ab = b2a, bc = c2b, ca = a2c〉
It’s trivial.
The sad realityDetermining whether a presentation gives the trivial group is undecidable.
The Word Problem (Dehn, 1911)Fix a finitely generated group G = 〈A〉. Determine whether a word in A istrivial in G.
Theorem (Novikov, 1954)There are groups for which the Word Problem is undecidable.
![Page 78: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/78.jpg)
The problem with presentations
Tell me something about this group
〈a, b, c | ab = b2a, bc = c2b, ca = a2c〉
It’s trivial.
The sad realityDetermining whether a presentation gives the trivial group is undecidable.
The Word Problem (Dehn, 1911)Fix a finitely generated group G = 〈A〉. Determine whether a word in A istrivial in G.
Theorem (Novikov, 1954)There are groups for which the Word Problem is undecidable.
![Page 79: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/79.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉
abaaabbaab
![Page 80: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/80.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉
abaaabbaab
![Page 81: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/81.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉
abaaabbaab
![Page 82: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/82.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉
abaaabbaab
![Page 83: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/83.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 84: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/84.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
11
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 85: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/85.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
a
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 86: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/86.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
ab
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 87: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/87.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
b
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 88: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/88.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
a3b
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 89: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/89.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
a2b
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 90: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/90.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
a2
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 91: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/91.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
a2b
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 92: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/92.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
ab
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 93: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/93.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
b
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 94: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/94.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
11
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 95: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/95.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
11
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 96: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/96.jpg)
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 97: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/97.jpg)
Theorem
(Anisimov, 1971)
A group G is finite ⇐⇒ word problem for G is solvable by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 98: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/98.jpg)
Theorem (Anisimov, 1971)A group G is finite ⇐⇒ word problem for G is solvable by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
![Page 99: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/99.jpg)
TheoremA group G is free =⇒ word problem for G is solvable
by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 100: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/100.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 101: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/101.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 102: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/102.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 103: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/103.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉
ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 104: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/104.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 105: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/105.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 106: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/106.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 107: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/107.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 108: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/108.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 109: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/109.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 110: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/110.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 111: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/111.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 112: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/112.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 113: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/113.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 114: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/114.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 115: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/115.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 116: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/116.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 117: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/117.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 118: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/118.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
![Page 119: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/119.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free =⇒ word problem for G is solvableby a push down automaton
![Page 120: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/120.jpg)
And so our scene must to the battle fly,
Where – O for pity! – we shall much disgrace
With four or five most vile and ragged foils,
Right ill-disposed in brawl ridiculous,
The name of Agincourt. Yet sit and see,
Minding true things by what their mockeries be.
– Henry V (Act IV, prologue)
![Page 121: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/121.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free =⇒ word problem for G is solvableby a push down automaton
![Page 122: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/122.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free ⇐⇒ word problem for G is solvableby a push down automaton
![Page 123: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/123.jpg)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem (Muller–Schupp, 1983)A group G is virtually free ⇐⇒ word problem for G is solvable
by a push down automaton
![Page 124: What is a group?sh15083/talks/2018.11.02_ps.pdf · Permutation group h(1 2 3 4),(1 4)(2 3)iRepresentation theory ˝ 0 1 1 0 , 1 0 0 1 ˛ Combinatorial group theory h a,bj4 = b2 =](https://reader034.vdocuments.us/reader034/viewer/2022050111/5f487b89ab6e9272042f0733/html5/thumbnails/124.jpg)
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab