Page 112 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 112
3.4 Some extension theory
gap> SplitExtension( G, A );;
gap> StructureDescription(last);
"C2 x C2 x D8"
gap> ext := Extensions( G, A );;
gap> Length(ext);
64
gap> DuplicateFreeList(List(ext, IdGroup));
[ [ 32, 46 ], [ 32, 40 ], [ 32, 22 ], [ 32, 39 ], [ 32, 9 ], [ 32, 23 ],
[ 32, 13 ], [ 32, 41 ], [ 32, 10 ], [ 32, 2 ], [ 32, 14 ] ]
Here note that the notation C4 : C4 means that the group in question is a semidi-
rect product of C4 by C4. The command TwoCocycles(G, A) returns a list of vectors
over the field underlying A, and the additive group generated by these vectors is the
Z 2(G , A). There is also a command TwoCohomology(G, A) that returns a record defin-
ing the second cohomology group as factor space of the vector space of cocycles by the
subspace of coboundaries. We refer to GAP’s manual for further details.
gap> z2 := AdditiveGroupByGenerators(co);;
gap> Length(Elements(z2));
256
gap> h2 := TwoCohomology(G, A);;
gap> h2.cohom;
8 over GF(2)> -> ( GF(2)^6 )>
gap> dimensionZ2 := Dimension(Source(h2.cohom));
8
gap> dimensionB2 := Dimension(Kernel(h2.cohom));
2
gap> dimensionH2 := Dimension(Image(h2.cohom));
6
The last line tells us that H 2(G , A) ∼= C26.
3.4.4 The Schur-Zassenhaus theorem
Let A and G be groups. We say that an extension of A by G splits if it is a semidirect
product.
Theorem 3.4.10 Suppose that A and G are finite groups satisfying gcd(|A|, |G |) = 1. Then
every extension of A by G splits.
We will only prove this result in the case when A is abelian. In this form, the result
was originally due to Schur. Zassenhaus improved it by showing that it suffices to assume
that one of A or G is solvable. On the other hand, Feit-Thompson’s Odd Order Theorem
shows that this assumption is redundant.
PROOF.[Proof of Theorem 3.4.10 when A is abelian] Let m = |A| and n = |G |. Let φ : G ×
G → A be a 2-cocycle representing an extension of A by G , and let χ : G → Aut(A) be
the homomorphism that induces the corresponding G -module structure on A. We claim
gap> SplitExtension( G, A );;
gap> StructureDescription(last);
"C2 x C2 x D8"
gap> ext := Extensions( G, A );;
gap> Length(ext);
64
gap> DuplicateFreeList(List(ext, IdGroup));
[ [ 32, 46 ], [ 32, 40 ], [ 32, 22 ], [ 32, 39 ], [ 32, 9 ], [ 32, 23 ],
[ 32, 13 ], [ 32, 41 ], [ 32, 10 ], [ 32, 2 ], [ 32, 14 ] ]
Here note that the notation C4 : C4 means that the group in question is a semidi-
rect product of C4 by C4. The command TwoCocycles(G, A) returns a list of vectors
over the field underlying A, and the additive group generated by these vectors is the
Z 2(G , A). There is also a command TwoCohomology(G, A) that returns a record defin-
ing the second cohomology group as factor space of the vector space of cocycles by the
subspace of coboundaries. We refer to GAP’s manual for further details.
gap> z2 := AdditiveGroupByGenerators(co);;
gap> Length(Elements(z2));
256
gap> h2 := TwoCohomology(G, A);;
gap> h2.cohom;
gap> dimensionZ2 := Dimension(Source(h2.cohom));
8
gap> dimensionB2 := Dimension(Kernel(h2.cohom));
2
gap> dimensionH2 := Dimension(Image(h2.cohom));
6
The last line tells us that H 2(G , A) ∼= C26.
3.4.4 The Schur-Zassenhaus theorem
Let A and G be groups. We say that an extension of A by G splits if it is a semidirect
product.
Theorem 3.4.10 Suppose that A and G are finite groups satisfying gcd(|A|, |G |) = 1. Then
every extension of A by G splits.
We will only prove this result in the case when A is abelian. In this form, the result
was originally due to Schur. Zassenhaus improved it by showing that it suffices to assume
that one of A or G is solvable. On the other hand, Feit-Thompson’s Odd Order Theorem
shows that this assumption is redundant.
PROOF.[Proof of Theorem 3.4.10 when A is abelian] Let m = |A| and n = |G |. Let φ : G ×
G → A be a 2-cocycle representing an extension of A by G , and let χ : G → Aut(A) be
the homomorphism that induces the corresponding G -module structure on A. We claim
