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Sporadic Symmetry(The Remarkable Behaviour of the Mathieu Groups)
Scott Harper
MT5999 Presentation
17th April 2015
Scott Harper Sporadic Symmetry 17th April 2015 1 / 9
Sporadic Symmetry(The Remarkable Behaviour of the Mathieu Groups)
Scott Harper
MT5999 Presentation
17th April 2015
Scott Harper Sporadic Symmetry 17th April 2015 2 / 9
It is a remarkable fact that ...
From Sphere Packings, Lattices and Groups by Conway and Sloane,
“At one point while working on this book we evenconsidered adopting a special abbreviation for ‘It is aremarkable fact that’ since this phrase seemed to occurso often. But in fact we have tried to avoid such phrasesand to maintain a scholarly decorum of language.”
Theorem
The symmetric group Sn has a non-trivial outer automorphism ifand only if n = 6.
Scott Harper Sporadic Symmetry 17th April 2015 3 / 9
It is a remarkable fact that ...
From Sphere Packings, Lattices and Groups by Conway and Sloane,
“At one point while working on this book we evenconsidered adopting a special abbreviation for ‘It is aremarkable fact that’ since this phrase seemed to occurso often. But in fact we have tried to avoid such phrasesand to maintain a scholarly decorum of language.”
Theorem
The symmetric group Sn has a non-trivial outer automorphism ifand only if n = 6.
Scott Harper Sporadic Symmetry 17th April 2015 3 / 9
Transitivity
Definition
Let G be a group acting on a set X . We sayG acts
transitively if for all x , y ∈ X thereexists g ∈ G such that xg = y ;
k-transitively if for all sequences ofdistinct points(x1, . . . , xk), (y1, . . . , yk) ∈ X k thereexists g ∈ G such that xig = yi ;
sharply k-transitively if the above g isthe unique such element.
4
1
3
2
Scott Harper Sporadic Symmetry 17th April 2015 4 / 9
Transitivity
Definition
Let G be a group acting on a set X . We sayG acts
transitively if for all x , y ∈ X thereexists g ∈ G such that xg = y ;
k-transitively if for all sequences ofdistinct points(x1, . . . , xk), (y1, . . . , yk) ∈ X k thereexists g ∈ G such that xig = yi ;
sharply k-transitively if the above g isthe unique such element.
4
1
3
2
Scott Harper Sporadic Symmetry 17th April 2015 4 / 9
Transitivity
Definition
Let G be a group acting on a set X . We sayG acts
transitively if for all x , y ∈ X thereexists g ∈ G such that xg = y ;
k-transitively if for all sequences ofdistinct points(x1, . . . , xk), (y1, . . . , yk) ∈ X k thereexists g ∈ G such that xig = yi ;
sharply k-transitively if the above g isthe unique such element.
4
1
3
2
Scott Harper Sporadic Symmetry 17th April 2015 4 / 9
Transitivity
Definition
Let G be a group acting on a set X . We sayG acts
transitively if for all x , y ∈ X thereexists g ∈ G such that xg = y ;
k-transitively if for all sequences ofdistinct points(x1, . . . , xk), (y1, . . . , yk) ∈ X k thereexists g ∈ G such that xig = yi ;
sharply k-transitively if the above g isthe unique such element.
4
1
3
2
Scott Harper Sporadic Symmetry 17th April 2015 4 / 9
Transitivity
Definition
Let G be a group acting on a set X . We sayG acts
transitively if for all x , y ∈ X thereexists g ∈ G such that xg = y ;
k-transitively if for all sequences ofdistinct points(x1, . . . , xk), (y1, . . . , yk) ∈ X k thereexists g ∈ G such that xig = yi ;
sharply k-transitively if the above g isthe unique such element.
4
1
3
2
Scott Harper Sporadic Symmetry 17th April 2015 4 / 9
Simplicity
Theorem (The Classification of Finite Simple Groups)
Every finite simple group is isomorphic to one of the following groups:
a cyclic group of prime order;
an alternating group of degree at least 5;
a simple group in one of the 16 families of groups of Lie type;
one of the 26 sporadic simple groups.
Theorem
Let G be a group acting k-transitively on a set. If G is not a symmetric oralternating group then k ≤ 5.
Moreover,
if k = 5 then G is the Mathieu group M12 or M24;
if k = 4 then G is the Mathieu group M11 or M23.
Note: M22 acts 3-transitively.
Scott Harper Sporadic Symmetry 17th April 2015 5 / 9
Simplicity
Theorem (The Classification of Finite Simple Groups)
Every finite simple group is isomorphic to one of the following groups:
a cyclic group of prime order;
an alternating group of degree at least 5;
a simple group in one of the 16 families of groups of Lie type;
one of the 26 sporadic simple groups.
Theorem
Let G be a group acting k-transitively on a set. If G is not a symmetric oralternating group then k ≤ 5.
Moreover,
if k = 5 then G is the Mathieu group M12 or M24;
if k = 4 then G is the Mathieu group M11 or M23.
Note: M22 acts 3-transitively.
Scott Harper Sporadic Symmetry 17th April 2015 5 / 9
Simplicity
Theorem (The Classification of Finite Simple Groups)
Every finite simple group is isomorphic to one of the following groups:
a cyclic group of prime order;
an alternating group of degree at least 5;
a simple group in one of the 16 families of groups of Lie type;
one of the 26 sporadic simple groups.
Theorem
Let G be a group acting k-transitively on a set. If G is not a symmetric oralternating group then k ≤ 5.
Moreover,
if k = 5 then G is the Mathieu group M12 or M24;
if k = 4 then G is the Mathieu group M11 or M23.
Note: M22 acts 3-transitively.
Scott Harper Sporadic Symmetry 17th April 2015 5 / 9
Simplicity
Theorem (The Classification of Finite Simple Groups)
Every finite simple group is isomorphic to one of the following groups:
a cyclic group of prime order;
an alternating group of degree at least 5;
a simple group in one of the 16 families of groups of Lie type;
one of the 26 sporadic simple groups.
Theorem
Let G be a group acting k-transitively on a set. If G is not a symmetric oralternating group then k ≤ 5. Moreover,
if k = 5 then G is the Mathieu group M12 or M24;
if k = 4 then G is the Mathieu group M11 or M23.
Note: M22 acts 3-transitively.
Scott Harper Sporadic Symmetry 17th April 2015 5 / 9
Simplicity
Theorem (The Classification of Finite Simple Groups)
Every finite simple group is isomorphic to one of the following groups:
a cyclic group of prime order;
an alternating group of degree at least 5;
a simple group in one of the 16 families of groups of Lie type;
one of the 26 sporadic simple groups.
Theorem
Let G be a group acting k-transitively on a set. If G is not a symmetric oralternating group then k ≤ 5. Moreover,
if k = 5 then G is the Mathieu group M12 or M24;
if k = 4 then G is the Mathieu group M11 or M23.
Note: M22 acts 3-transitively.
Scott Harper Sporadic Symmetry 17th April 2015 5 / 9
Steiner Systems
The affine plane AG2(3)
Definition
An S(t, k , v) Steiner system is aset X of v points and a set ofk-element subsets of X such thatany t points lie in a unique suchsubset.
Example
The affine plane AG2(3) is anS(2, 3, 9) Steiner system.
Scott Harper Sporadic Symmetry 17th April 2015 6 / 9
Steiner Systems
The affine plane AG2(3)
Definition
An S(t, k , v) Steiner system is aset X of v points and a set ofk-element subsets of X such thatany t points lie in a unique suchsubset.
Example
The affine plane AG2(3) is anS(2, 3, 9) Steiner system.
Scott Harper Sporadic Symmetry 17th April 2015 6 / 9
Steiner Systems
The affine plane AG2(3)
Definition
An S(t, k , v) Steiner system is aset X of v points and a set ofk-element subsets of X such thatany t points lie in a unique suchsubset.
Example
The affine plane AG2(3) is anS(2, 3, 9) Steiner system.
Scott Harper Sporadic Symmetry 17th April 2015 6 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x ≤ Aut(W11) =: M11
S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3)
(M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x ≤ Aut(W11) =: M11
S(3, 4, 10)
W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3)
(M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11)
W11 (M12)x ≤ Aut(W11) =: M11
S(3, 4, 10)
W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3)
(M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12)
W12 M12 := Aut(W12)
S(4, 5, 11)
W11 (M12)x ≤ Aut(W11) =: M11
S(3, 4, 10)
W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3)
(M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12
M12 := Aut(W12)
S(4, 5, 11) W11
(M12)x ≤ Aut(W11) =: M11
S(3, 4, 10) W10
(M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3)
(M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12
M12 := Aut(W12)
S(4, 5, 11) W11
(M12)x ≤ Aut(W11) =: M11
S(3, 4, 10) W10
(M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3)
(M12)xyz ≤
Aut(AG2(3))
=
=
H ≤
AGL2(3)
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12
M12 := Aut(W12)
S(4, 5, 11) W11
(M12)x ≤ Aut(W11) =: M11
S(3, 4, 10) W10
(M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3)
(M12)xyz ≤
Aut(AG2(3))
=
=
H ≤ AGL2(3)
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11
(M12)x ≤
Aut(W11)
=: M11
S(3, 4, 10) W10
(M12)xy ≤
Aut(W10)
S(2, 3, 9) AG2(3)
(M12)xyz ≤
Aut(AG2(3))
=
=
H ≤ AGL2(3)
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x ≤ Aut(W11)
=: M11
S(3, 4, 10) W10
(M12)xy ≤
Aut(W10)
S(2, 3, 9) AG2(3)
(M12)xyz ≤
Aut(AG2(3))
=
=
H ≤ AGL2(3)
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x ≤ Aut(W11)
=: M11
S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3)
(M12)xyz ≤
Aut(AG2(3))
=
=
H ≤ AGL2(3)
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x ≤ Aut(W11)
=: M11
S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))
=
=
H ≤ AGL2(3)
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x ≤ Aut(W11)
=: M11
S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x ≤ Aut(W11)
=: M11
S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Theorem
Suppose that
- G acts transitively on X
- Gx acts k-transitively on X \ {x}, for some x ∈ X .
Then G acts (k + 1)-transitively on X .
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x ≤ Aut(W11)
=: M11
S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Theorem
Suppose that
- G acts transitively on X
- Gx acts sharply k-transitively on X \ {x}, for some x ∈ X .
Then G acts sharply (k + 1)-transitively on X .
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x ≤ Aut(W11)
=: M11
S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Theorem
Suppose that
- G acts transitively on X
- Gx acts sharply k-transitively on X \ {x}, for some x ∈ X .
Then G acts sharply (k + 1)-transitively on X .
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x ≤ Aut(W11)
=: M11
S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Theorem
Suppose that
- G acts transitively on X
- Gx acts sharply k-transitively on X \ {x}, for some x ∈ X .
Then G acts sharply (k + 1)-transitively on X .
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x ≤ Aut(W11)
=: M11
S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Theorem
Suppose that
- G acts transitively on X
- Gx acts sharply k-transitively on X \ {x}, for some x ∈ X .
Then G acts sharply (k + 1)-transitively on X .
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x = Aut(W11)
=: M11
S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Theorem
Suppose that
- G acts transitively on X
- Gx acts sharply k-transitively on X \ {x}, for some x ∈ X .
Then G acts sharply (k + 1)-transitively on X .
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x = Aut(W11) =: M11
S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Theorem
Suppose that
- G acts transitively on X
- Gx acts sharply k-transitively on X \ {x}, for some x ∈ X .
Then G acts sharply (k + 1)-transitively on X .
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x = Aut(W11) =: M11
S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Theorem
Suppose that
- G acts transitively on X
- Gx acts sharply k-transitively on X \ {x}, for some x ∈ X .
Then G acts sharply (k + 1)-transitively on X .
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x = Aut(W11) =: M11
S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Theorem
Suppose that
- G acts k-transitively on X for k ≥ 3 and |X | not equal to 3 or 2n
- Gx is simple, for some x ∈ X .
Then G is simple.
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
Witt Geometries and Mathieu Groups
S(5, 6, 12) W12 M12 := Aut(W12)
S(4, 5, 11) W11 (M12)x = Aut(W11) =: M11
S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)
S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))
= =
H ≤ AGL2(3)
Theorem
Suppose that
- G acts k-transitively on X for k ≥ 3 and |X | not equal to 3 or 2n
- Gx is simple, for some x ∈ X .
Then G is simple.
Scott Harper Sporadic Symmetry 17th April 2015 7 / 9
M12 acting on W12
φ : g 7→ g |B ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6 SB SC S6
G
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6 SB SC S6
G
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B ψ : g 7→ g |C
B
C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6 SB SC S6
G
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6 SB SC S6
G
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6 SB SC S6
G
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B
ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6
SB
SC S6
Gφ
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B
ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6
SB
SC S6
Gφ
The map φ is a homomorphism,
aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B
ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6
SB
SC S6
Gφ
The map φ is a homomorphism, aninjection
and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B
ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6
SB
SC S6
Gφ
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6
SB SC
S6
Gφ ψ
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6
SB SC
S6
Gφ ψ
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6
SB SC
S6
Gφ ψ
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6
SB SC
S6
Gφ ψ
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6
SB SC
S6
Gφ ψ
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6 SB SC
S6
G
β
φ ψ
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6 SB SC S6
G
β γ
φ ψ
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6 SB SC S6
G
β γ
φ ψ
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6 SB SC S6
G
β γ
φ ψ
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
M12 acting on W12
φ : g 7→ g |B ψ : g 7→ g |C
B C
123456
b1b2b3b4b5b6
c1c2c3c4c5c6
123456
S6 SB SC S6
G
β γ
φ ψ
The map φ is a homomorphism, aninjection and a surjection.
Fact: φ−1ψ : SB → SC is anisomorphism.
Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)
Fact: βφ−1ψγ : S6 → S6 isan isomorphism.
Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)
So βφ−1ψγ is a non-trivialouter automorphism of S6.
Scott Harper Sporadic Symmetry 17th April 2015 8 / 9
Mathieu Groups
“These apparently sporadic simple groups would probably repay a closerexamination than they have yet received.”
(William Burnside)
Any questions?
Scott Harper Sporadic Symmetry 17th April 2015 9 / 9
Mathieu Groups
“These apparently sporadic simple groups would probably repay a closerexamination than they have yet received.”
(William Burnside)
Any questions?
Scott Harper Sporadic Symmetry 17th April 2015 9 / 9