a ne complete g -setsical/presentations/pongracz.pdf · de nition a ne completeness an algebra aisa...
TRANSCRIPT
![Page 1: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/1.jpg)
Ane complete G -sets
András Pongrácz, Gábor Horváth, Peter Mayr
ELTE Budapest, Hungary
Prague, June 2010.
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 2: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/2.jpg)
Grätzer
Describe the ane complete algebras.
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 3: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/3.jpg)
Denition
Ane completeness
An algebra A is ane complete if every compatible function is a
polynomial on A.
Denition
f : Ak → A compatible : for all θ ∈ Con(A) we have
(a,b) ∈ θ ⇒ (f (a), f (b)) ∈ θ.
constants, fundamental operations
polynomials
k-ane completeness
An algebra A is k-ane coplete if every compatible function of arity at
most k is a polynomial on A.
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 4: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/4.jpg)
Denition
Ane completeness
An algebra A is ane complete if every compatible function is a
polynomial on A.
Denition
f : Ak → A compatible : for all θ ∈ Con(A) we have
(a,b) ∈ θ ⇒ (f (a), f (b)) ∈ θ.
constants, fundamental operations
polynomials
k-ane completeness
An algebra A is k-ane coplete if every compatible function of arity at
most k is a polynomial on A.
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 5: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/5.jpg)
Denition
Ane completeness
An algebra A is ane complete if every compatible function is a
polynomial on A.
Denition
f : Ak → A compatible : for all θ ∈ Con(A) we have
(a,b) ∈ θ ⇒ (f (a), f (b)) ∈ θ.constants, fundamental operations
polynomials
k-ane completeness
An algebra A is k-ane coplete if every compatible function of arity at
most k is a polynomial on A.
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 6: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/6.jpg)
Denition
Ane completeness
An algebra A is ane complete if every compatible function is a
polynomial on A.
Denition
f : Ak → A compatible : for all θ ∈ Con(A) we have
(a,b) ∈ θ ⇒ (f (a), f (b)) ∈ θ.constants, fundamental operations
polynomials
k-ane completeness
An algebra A is k-ane coplete if every compatible function of arity at
most k is a polynomial on A.
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 7: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/7.jpg)
Denition
Ane completeness
An algebra A is ane complete if every compatible function is a
polynomial on A.
Denition
f : Ak → A compatible : for all θ ∈ Con(A) we have
(a,b) ∈ θ ⇒ (f (a), f (b)) ∈ θ.constants, fundamental operations
polynomials
k-ane completeness
An algebra A is k-ane coplete if every compatible function of arity at
most k is a polynomial on A.
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 8: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/8.jpg)
Examples
Fields, Boolean algebras
Finite elds, 2-element Boolean algebra: every function is a polynomial.
These are ane complete.
Zp
every function is compatible
polynomials:n∑i=1
aixi + b
⇒ not ane complete
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 9: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/9.jpg)
Examples
Fields, Boolean algebras
Finite elds, 2-element Boolean algebra: every function is a polynomial.
These are ane complete.
Zp
every function is compatible
polynomials:n∑i=1
aixi + b
⇒ not ane complete
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 10: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/10.jpg)
Examples
Fields, Boolean algebras
Finite elds, 2-element Boolean algebra: every function is a polynomial.
These are ane complete.
Zp
every function is compatible
polynomials:n∑i=1
aixi + b
⇒ not ane complete
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 11: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/11.jpg)
Examples
Fields, Boolean algebras
Finite elds, 2-element Boolean algebra: every function is a polynomial.
These are ane complete.
Zp
every function is compatible
polynomials:n∑i=1
aixi + b
⇒ not ane complete
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 12: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/12.jpg)
Examples
Fields, Boolean algebras
Finite elds, 2-element Boolean algebra: every function is a polynomial.
These are ane complete.
Zp
every function is compatible
polynomials:n∑i=1
aixi + b
⇒ not ane complete
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 13: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/13.jpg)
Grätzer
Describe the ane complete algebras.
First signicant results:
Boolean algebras
Bounded distributive lattices: not containing proper Boolean intervals
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 14: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/14.jpg)
Grätzer
Describe the ane complete algebras.
First signicant results:
Boolean algebras
Bounded distributive lattices: not containing proper Boolean intervals
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 15: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/15.jpg)
Grätzer
Describe the ane complete algebras.
First signicant results:
Boolean algebras
Bounded distributive lattices: not containing proper Boolean intervals
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 16: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/16.jpg)
Grätzer
Describe the ane complete algebras.
First signicant results:
Boolean algebras
Bounded distributive lattices: not containing proper Boolean intervals
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 17: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/17.jpg)
Some known results
Abelian groups: K. Kaarli
Semilattices: K. Kaarli, L. Márki, E. T. Schmidt
Vector spaces: H. Werner
Distributive lattices: M. Plos c ica
Stone algebras: M. Haviar, M. Plos c ica
Kleene algebras: M. Haviar, K. Kaarli, M. Plos c ica
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 18: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/18.jpg)
Denition
G-set
(Ω,G ) pair
underlying set: Ω
type: for all g ∈ G a unary operation is given:
mg : α 7→ gα
Transitive group actions
S ≤ G , Ω: cosets of S
mg (hS) = ghS
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 19: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/19.jpg)
Denition
G-set
(Ω,G ) pair
underlying set: Ω
type: for all g ∈ G a unary operation is given:
mg : α 7→ gα
Transitive group actions
S ≤ G , Ω: cosets of S
mg (hS) = ghS
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 20: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/20.jpg)
Denition
G-set
(Ω,G ) pair
underlying set: Ω
type: for all g ∈ G a unary operation is given:
mg : α 7→ gα
Transitive group actions
S ≤ G , Ω: cosets of S
mg (hS) = ghS
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 21: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/21.jpg)
Denition
G-set
(Ω,G ) pair
underlying set: Ω
type: for all g ∈ G a unary operation is given:
mg : α 7→ gα
Transitive group actions
S ≤ G , Ω: cosets of S
mg (hS) = ghS
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 22: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/22.jpg)
Denition
G-set
(Ω,G ) pair
underlying set: Ω
type: for all g ∈ G a unary operation is given:
mg : α 7→ gα
Transitive group actions
S ≤ G , Ω: cosets of S
mg (hS) = ghS
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 23: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/23.jpg)
Denition
G-set
(Ω,G ) pair
underlying set: Ω
type: for all g ∈ G a unary operation is given:
mg : α 7→ gα
Transitive group actions
S ≤ G , Ω: cosets of S
mg (hS) = ghS
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 24: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/24.jpg)
Denition
G-set
(Ω,G ) pair
underlying set: Ω
type: for all g ∈ G a unary operation is given:
mg : α 7→ gα
Transitive group actions
S ≤ G , Ω: cosets of S
mg (hS) = ghS
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 25: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/25.jpg)
Transitive group actions
Congruences
S ≤ H ≤ G
ρH = (aS , bS) ∈ Ω× Ω|a, b ∈ G , aH = bH
Congruences ! H|S ≤ H ≤ GCon((Ω,G )) = [S ,G ] ≤ L(G )
Regular G-sets
Con(R(G )) = ρH |H ≤ G ∼= L(G ) (subgroup lattice)
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 26: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/26.jpg)
Transitive group actions
Congruences
S ≤ H ≤ G
ρH = (aS , bS) ∈ Ω× Ω|a, b ∈ G , aH = bH
Congruences ! H|S ≤ H ≤ GCon((Ω,G )) = [S ,G ] ≤ L(G )
Regular G-sets
Con(R(G )) = ρH |H ≤ G ∼= L(G ) (subgroup lattice)
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 27: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/27.jpg)
Transitive group actions
Congruences
S ≤ H ≤ G
ρH = (aS , bS) ∈ Ω× Ω|a, b ∈ G , aH = bH
Congruences ! H|S ≤ H ≤ GCon((Ω,G )) = [S ,G ] ≤ L(G )
Regular G-sets
Con(R(G )) = ρH |H ≤ G ∼= L(G ) (subgroup lattice)
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 28: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/28.jpg)
Transitive group actions
Congruences
S ≤ H ≤ G
ρH = (aS , bS) ∈ Ω× Ω|a, b ∈ G , aH = bH
Congruences ! H|S ≤ H ≤ GCon((Ω,G )) = [S ,G ] ≤ L(G )
Regular G-sets
Con(R(G )) = ρH |H ≤ G ∼= L(G ) (subgroup lattice)
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 29: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/29.jpg)
Transitive group actions
Congruences
S ≤ H ≤ G
ρH = (aS , bS) ∈ Ω× Ω|a, b ∈ G , aH = bH
Congruences ! H|S ≤ H ≤ G
Con((Ω,G )) = [S ,G ] ≤ L(G )
Regular G-sets
Con(R(G )) = ρH |H ≤ G ∼= L(G ) (subgroup lattice)
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 30: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/30.jpg)
Transitive group actions
Congruences
S ≤ H ≤ G
ρH = (aS , bS) ∈ Ω× Ω|a, b ∈ G , aH = bH
Congruences ! H|S ≤ H ≤ GCon((Ω,G )) = [S ,G ] ≤ L(G )
Regular G-sets
Con(R(G )) = ρH |H ≤ G ∼= L(G ) (subgroup lattice)
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 31: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/31.jpg)
Transitive group actions
Congruences
S ≤ H ≤ G
ρH = (aS , bS) ∈ Ω× Ω|a, b ∈ G , aH = bH
Congruences ! H|S ≤ H ≤ GCon((Ω,G )) = [S ,G ] ≤ L(G )
Regular G-sets
Con(R(G )) = ρH |H ≤ G ∼= L(G ) (subgroup lattice)
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 32: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/32.jpg)
Regular actions
Unary compatible functions
R(G ): regular permutation representation of G
f compatible on R(G ): for all H ≤ G and a, b ∈ G
aH = bH ⇒ f (a)H = f (b)H
constants x 7→ g , g ∈ G
left translations x 7→ gx , g ∈ G
These are the only unary polynomial functions on R(G ).
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 33: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/33.jpg)
Regular actions
Unary compatible functions
R(G ): regular permutation representation of G
f compatible on R(G ): for all H ≤ G and a, b ∈ G
aH = bH ⇒ f (a)H = f (b)H
constants x 7→ g , g ∈ G
left translations x 7→ gx , g ∈ G
These are the only unary polynomial functions on R(G ).
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 34: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/34.jpg)
Regular actions
Unary compatible functions
R(G ): regular permutation representation of G
f compatible on R(G ): for all H ≤ G and a, b ∈ G
aH = bH ⇒ f (a)H = f (b)H
constants x 7→ g , g ∈ G
left translations x 7→ gx , g ∈ G
These are the only unary polynomial functions on R(G ).
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 35: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/35.jpg)
Regular actions
Unary compatible functions
R(G ): regular permutation representation of G
f compatible on R(G ): for all H ≤ G and a, b ∈ G
aH = bH ⇒ f (a)H = f (b)H
constants x 7→ g , g ∈ G
left translations x 7→ gx , g ∈ G
These are the only unary polynomial functions on R(G ).
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 36: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/36.jpg)
Regular actions
Unary compatible functions
R(G ): regular permutation representation of G
f compatible on R(G ): for all H ≤ G and a, b ∈ G
aH = bH ⇒ f (a)H = f (b)H
constants x 7→ g , g ∈ G
left translations x 7→ gx , g ∈ G
These are the only unary polynomial functions on R(G ).
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 37: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/37.jpg)
Regular actions
Unary compatible functions
R(G ): regular permutation representation of G
f compatible on R(G ): for all H ≤ G and a, b ∈ G
aH = bH ⇒ f (a)H = f (b)H
constants x 7→ g , g ∈ G
left translations x 7→ gx , g ∈ G
These are the only unary polynomial functions on R(G ).
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Regular actions
Unary compatible functions
R(G ): regular permutation representation of G
f compatible on R(G ): for all H ≤ G and a, b ∈ G
aH = bH ⇒ f (a)H = f (b)H
constants x 7→ g , g ∈ G
left translations x 7→ gx , g ∈ G
These are the only unary polynomial functions on R(G ).
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1-ane completeness
Theorem
Pálfy (1984.): Classication of minimal algebras.
Corollary
(Ω,G ) 1-ane complete ⇒ ane complete, except:
|Ω| = 2
There exists a division ring D and a vector space DV such that
Ω =D V and G = x 7→ dx + v |d ∈ D, v ∈ V
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1-ane completeness
Theorem
Pálfy (1984.): Classication of minimal algebras.
Corollary
(Ω,G ) 1-ane complete ⇒ ane complete, except:
|Ω| = 2
There exists a division ring D and a vector space DV such that
Ω =D V and G = x 7→ dx + v |d ∈ D, v ∈ V
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1-ane completeness
Theorem
Pálfy (1984.): Classication of minimal algebras.
Corollary
(Ω,G ) 1-ane complete ⇒ ane complete, except:
|Ω| = 2
There exists a division ring D and a vector space DV such that
Ω =D V and G = x 7→ dx + v |d ∈ D, v ∈ V
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1-ane completeness
Theorem
Pálfy (1984.): Classication of minimal algebras.
Corollary
(Ω,G ) 1-ane complete ⇒ ane complete, except:
|Ω| = 2
There exists a division ring D and a vector space DV such that
Ω =D V and G = x 7→ dx + v |d ∈ D, v ∈ V
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Notation
t-completeness
A group G is t-complete if R(G ) is a 1-ane complete G -set.
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Necessary conditions
Proposition
Let G be an abelian group. Then G is not t-complete, except if G is an
elementary abelian 2-group.
Proposition
x 7→ x |G :Z(G)| (transfer) is compatible
G is t-complete ⇒ |G : Z (G )| is divisible by exp(G ).
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Necessary conditions
Proposition
Let G be an abelian group. Then G is not t-complete, except if G is an
elementary abelian 2-group.
Proposition
x 7→ x |G :Z(G)| (transfer) is compatible
G is t-complete ⇒ |G : Z (G )| is divisible by exp(G ).
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Necessary conditions
Lemma (Typical counterexample)
G = A× B , A,B are proper subgroups and gcd(|A|, |B|) = 1. Then G is
not t-complete.
Proof
gcd(|A|, |B|) = 1
⇒ subgroups of G are of the form C × D.
π : (a, b) 7→ (a, 1) is compatible
(a1, b1)C × D = (a2, b2)C × D
⇒ a1C = a2C
⇒ (a1, 1)C × D = (a2, 1)C × D
π is not a polynomial function
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Necessary conditions
Lemma (Typical counterexample)
G = A× B , A,B are proper subgroups and gcd(|A|, |B|) = 1. Then G is
not t-complete.
Proof
gcd(|A|, |B|) = 1
⇒ subgroups of G are of the form C × D.
π : (a, b) 7→ (a, 1) is compatible
(a1, b1)C × D = (a2, b2)C × D
⇒ a1C = a2C
⇒ (a1, 1)C × D = (a2, 1)C × D
π is not a polynomial function
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Necessary conditions
Lemma (Typical counterexample)
G = A× B , A,B are proper subgroups and gcd(|A|, |B|) = 1. Then G is
not t-complete.
Proof
gcd(|A|, |B|) = 1
⇒ subgroups of G are of the form C × D.
π : (a, b) 7→ (a, 1) is compatible
(a1, b1)C × D = (a2, b2)C × D
⇒ a1C = a2C
⇒ (a1, 1)C × D = (a2, 1)C × D
π is not a polynomial function
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Necessary conditions
Lemma (Typical counterexample)
G = A× B , A,B are proper subgroups and gcd(|A|, |B|) = 1. Then G is
not t-complete.
Proof
gcd(|A|, |B|) = 1 ⇒ subgroups of G are of the form C × D.
π : (a, b) 7→ (a, 1) is compatible
(a1, b1)C × D = (a2, b2)C × D
⇒ a1C = a2C
⇒ (a1, 1)C × D = (a2, 1)C × D
π is not a polynomial function
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Necessary conditions
Lemma (Typical counterexample)
G = A× B , A,B are proper subgroups and gcd(|A|, |B|) = 1. Then G is
not t-complete.
Proof
gcd(|A|, |B|) = 1 ⇒ subgroups of G are of the form C × D.
π : (a, b) 7→ (a, 1) is compatible
(a1, b1)C × D = (a2, b2)C × D
⇒ a1C = a2C
⇒ (a1, 1)C × D = (a2, 1)C × D
π is not a polynomial function
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Necessary conditions
Lemma (Typical counterexample)
G = A× B , A,B are proper subgroups and gcd(|A|, |B|) = 1. Then G is
not t-complete.
Proof
gcd(|A|, |B|) = 1 ⇒ subgroups of G are of the form C × D.
π : (a, b) 7→ (a, 1) is compatible
(a1, b1)C × D = (a2, b2)C × D
⇒ a1C = a2C
⇒ (a1, 1)C × D = (a2, 1)C × D
π is not a polynomial function
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Necessary conditions
Lemma (Typical counterexample)
G = A× B , A,B are proper subgroups and gcd(|A|, |B|) = 1. Then G is
not t-complete.
Proof
gcd(|A|, |B|) = 1 ⇒ subgroups of G are of the form C × D.
π : (a, b) 7→ (a, 1) is compatible
(a1, b1)C × D = (a2, b2)C × D
⇒ a1C = a2C
⇒ (a1, 1)C × D = (a2, 1)C × D
π is not a polynomial function
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Necessary conditions
Lemma (Typical counterexample)
G = A× B , A,B are proper subgroups and gcd(|A|, |B|) = 1. Then G is
not t-complete.
Proof
gcd(|A|, |B|) = 1 ⇒ subgroups of G are of the form C × D.
π : (a, b) 7→ (a, 1) is compatible
(a1, b1)C × D = (a2, b2)C × D
⇒ a1C = a2C
⇒ (a1, 1)C × D = (a2, 1)C × D
π is not a polynomial function
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Necessary conditions
Lemma (Typical counterexample)
G = A× B , A,B are proper subgroups and gcd(|A|, |B|) = 1. Then G is
not t-complete.
Proof
gcd(|A|, |B|) = 1 ⇒ subgroups of G are of the form C × D.
π : (a, b) 7→ (a, 1) is compatible
(a1, b1)C × D = (a2, b2)C × D
⇒ a1C = a2C
⇒ (a1, 1)C × D = (a2, 1)C × D
π is not a polynomial function
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Necessary conditions
Theorem
Let A,B ≤ G such that 〈A,B〉 = G . If A and B are t-complete then
exactly one of the following holds:
G is t-complete
G = A× B and gcd(|A|, |B|) = 1.
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Necessary conditions
Theorem
Let A,B ≤ G such that 〈A,B〉 = G . If A and B are t-complete then
exactly one of the following holds:
G is t-complete
G = A× B and gcd(|A|, |B|) = 1.
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Necessary conditions
Theorem
Let A,B ≤ G such that 〈A,B〉 = G . If A and B are t-complete then
exactly one of the following holds:
G is t-complete
G = A× B and gcd(|A|, |B|) = 1.
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Corollary
Denition
t(G ) := 〈H ≤ G | H is t-complete 〉.
Properties of t(G )
t(G ) char G .
t(G ) is the direct product of t-complete maximal subgroups.
t(G ) = 1 implies that G is odd and any two elements of prime order
commute.
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Corollary
Denition
t(G ) := 〈H ≤ G | H is t-complete 〉.
Properties of t(G )
t(G ) char G .
t(G ) is the direct product of t-complete maximal subgroups.
t(G ) = 1 implies that G is odd and any two elements of prime order
commute.
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Corollary
Denition
t(G ) := 〈H ≤ G | H is t-complete 〉.
Properties of t(G )
t(G ) char G .
t(G ) is the direct product of t-complete maximal subgroups.
t(G ) = 1 implies that G is odd and any two elements of prime order
commute.
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Corollary
Denition
t(G ) := 〈H ≤ G | H is t-complete 〉.
Properties of t(G )
t(G ) char G .
t(G ) is the direct product of t-complete maximal subgroups.
t(G ) = 1 implies that G is odd and any two elements of prime order
commute.
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Corollary
Denition
t(G ) := 〈H ≤ G | H is t-complete 〉.
Properties of t(G )
t(G ) char G .
t(G ) is the direct product of t-complete maximal subgroups.
t(G ) = 1 implies that G is odd and any two elements of prime order
commute.
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Homomorphic images
Denition
T (G ) :=⋂N / G | G/N is t-complete .
Properties of T (G )
T (G ) char G .
G/T (G ) is the subdirect product of pairwise coprime t-complete
groups.
For nite G : T (G ) = G implies that G is nilpotent of odd order.
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Homomorphic images
Denition
T (G ) :=⋂N / G | G/N is t-complete .
Properties of T (G )
T (G ) char G .
G/T (G ) is the subdirect product of pairwise coprime t-complete
groups.
For nite G : T (G ) = G implies that G is nilpotent of odd order.
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Homomorphic images
Denition
T (G ) :=⋂N / G | G/N is t-complete .
Properties of T (G )
T (G ) char G .
G/T (G ) is the subdirect product of pairwise coprime t-complete
groups.
For nite G : T (G ) = G implies that G is nilpotent of odd order.
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Homomorphic images
Denition
T (G ) :=⋂N / G | G/N is t-complete .
Properties of T (G )
T (G ) char G .
G/T (G ) is the subdirect product of pairwise coprime t-complete
groups.
For nite G : T (G ) = G implies that G is nilpotent of odd order.
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Homomorphic images
Denition
T (G ) :=⋂N / G | G/N is t-complete .
Properties of T (G )
T (G ) char G .
G/T (G ) is the subdirect product of pairwise coprime t-complete
groups.
For nite G : T (G ) = G implies that G is nilpotent of odd order.
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t-complete groups
Theorem: Mayr, Horváth, Szabó, Pongrácz
The following groups are t-complete:
Nonabelian p-groups of exponent p.
Frobenius groups.
Dihedral groups, Sn (for all n), An (n ≥ 4), nite nonabelian simple
groups.
Finite nonabelian groups generated by their elements of prime order
that do not have the form G = A× B with gcd(|A|, |B|) = 1.
Groups all of whose minimal normal subgroups are nonabelian.
SL(n,F ) ≤ G ≤ GL(n,F ) (n ≥ 2)
Finite perfect groups.
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t-complete groups
Theorem: Mayr, Horváth, Szabó, Pongrácz
The following groups are t-complete:
Nonabelian p-groups of exponent p.
Frobenius groups.
Dihedral groups, Sn (for all n), An (n ≥ 4), nite nonabelian simple
groups.
Finite nonabelian groups generated by their elements of prime order
that do not have the form G = A× B with gcd(|A|, |B|) = 1.
Groups all of whose minimal normal subgroups are nonabelian.
SL(n,F ) ≤ G ≤ GL(n,F ) (n ≥ 2)
Finite perfect groups.
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t-complete groups
Theorem: Mayr, Horváth, Szabó, Pongrácz
The following groups are t-complete:
Nonabelian p-groups of exponent p.
Frobenius groups.
Dihedral groups, Sn (for all n), An (n ≥ 4), nite nonabelian simple
groups.
Finite nonabelian groups generated by their elements of prime order
that do not have the form G = A× B with gcd(|A|, |B|) = 1.
Groups all of whose minimal normal subgroups are nonabelian.
SL(n,F ) ≤ G ≤ GL(n,F ) (n ≥ 2)
Finite perfect groups.
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 71: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/71.jpg)
t-complete groups
Theorem: Mayr, Horváth, Szabó, Pongrácz
The following groups are t-complete:
Nonabelian p-groups of exponent p.
Frobenius groups.
Dihedral groups, Sn (for all n), An (n ≥ 4), nite nonabelian simple
groups.
Finite nonabelian groups generated by their elements of prime order
that do not have the form G = A× B with gcd(|A|, |B|) = 1.
Groups all of whose minimal normal subgroups are nonabelian.
SL(n,F ) ≤ G ≤ GL(n,F ) (n ≥ 2)
Finite perfect groups.
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 72: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/72.jpg)
t-complete groups
Theorem: Mayr, Horváth, Szabó, Pongrácz
The following groups are t-complete:
Nonabelian p-groups of exponent p.
Frobenius groups.
Dihedral groups, Sn (for all n), An (n ≥ 4), nite nonabelian simple
groups.
Finite nonabelian groups generated by their elements of prime order
that do not have the form G = A× B with gcd(|A|, |B|) = 1.
Groups all of whose minimal normal subgroups are nonabelian.
SL(n,F ) ≤ G ≤ GL(n,F ) (n ≥ 2)
Finite perfect groups.
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 73: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/73.jpg)
t-complete groups
Theorem: Mayr, Horváth, Szabó, Pongrácz
The following groups are t-complete:
Nonabelian p-groups of exponent p.
Frobenius groups.
Dihedral groups, Sn (for all n), An (n ≥ 4), nite nonabelian simple
groups.
Finite nonabelian groups generated by their elements of prime order
that do not have the form G = A× B with gcd(|A|, |B|) = 1.
Groups all of whose minimal normal subgroups are nonabelian.
SL(n,F ) ≤ G ≤ GL(n,F ) (n ≥ 2)
Finite perfect groups.
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 74: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/74.jpg)
t-complete groups
Theorem: Mayr, Horváth, Szabó, Pongrácz
The following groups are t-complete:
Nonabelian p-groups of exponent p.
Frobenius groups.
Dihedral groups, Sn (for all n), An (n ≥ 4), nite nonabelian simple
groups.
Finite nonabelian groups generated by their elements of prime order
that do not have the form G = A× B with gcd(|A|, |B|) = 1.
Groups all of whose minimal normal subgroups are nonabelian.
SL(n,F ) ≤ G ≤ GL(n,F ) (n ≥ 2)
Finite perfect groups.
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 75: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/75.jpg)
t-complete groups
Theorem: Mayr, Horváth, Szabó, Pongrácz
The following groups are t-complete:
Nonabelian p-groups of exponent p.
Frobenius groups.
Dihedral groups, Sn (for all n), An (n ≥ 4), nite nonabelian simple
groups.
Finite nonabelian groups generated by their elements of prime order
that do not have the form G = A× B with gcd(|A|, |B|) = 1.
Groups all of whose minimal normal subgroups are nonabelian.
SL(n,F ) ≤ G ≤ GL(n,F ) (n ≥ 2)
Finite perfect groups.
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 76: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/76.jpg)
Problems
Subgroup lattices
What lattices occur as L(G )?
Describe t-complete groups.
t(G), T(G)
What are the properties of t(G ) and T (G )?
Characterize groups with t(G ) = 1.
Characterize groups with T (G ) = G .
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 77: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/77.jpg)
Problems
Subgroup lattices
What lattices occur as L(G )?
Describe t-complete groups.
t(G), T(G)
What are the properties of t(G ) and T (G )?
Characterize groups with t(G ) = 1.
Characterize groups with T (G ) = G .
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 78: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/78.jpg)
Problems
Subgroup lattices
What lattices occur as L(G )?
Describe t-complete groups.
t(G), T(G)
What are the properties of t(G ) and T (G )?
Characterize groups with t(G ) = 1.
Characterize groups with T (G ) = G .
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 79: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/79.jpg)
Problems
Subgroup lattices
What lattices occur as L(G )?
Describe t-complete groups.
t(G), T(G)
What are the properties of t(G ) and T (G )?
Characterize groups with t(G ) = 1.
Characterize groups with T (G ) = G .
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 80: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/80.jpg)
Problems
Subgroup lattices
What lattices occur as L(G )?
Describe t-complete groups.
t(G), T(G)
What are the properties of t(G ) and T (G )?
Characterize groups with t(G ) = 1.
Characterize groups with T (G ) = G .
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 81: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/81.jpg)
Problems
Subgroup lattices
What lattices occur as L(G )?
Describe t-complete groups.
t(G), T(G)
What are the properties of t(G ) and T (G )?
Characterize groups with t(G ) = 1.
Characterize groups with T (G ) = G .
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets
![Page 82: A ne complete G -setsical/presentations/pongracz.pdf · De nition A ne completeness An algebra Aisa ne complete if every compatible function is a polynomial on A. De nition f : A](https://reader031.vdocuments.us/reader031/viewer/2022022421/5a7fd6557f8b9a682c8bd1c6/html5/thumbnails/82.jpg)
Problems
Subgroup lattices
What lattices occur as L(G )?
Describe t-complete groups.
t(G), T(G)
What are the properties of t(G ) and T (G )?
Characterize groups with t(G ) = 1.
Characterize groups with T (G ) = G .
András Pongrácz, Gábor Horváth, Peter Mayr Ane complete G-sets