Central Extensions

We now describe functions to construct \(H^2(G,U)\) for a finite soluble group \(G\) and finite abelian group \(U\) (a trivial \(G\)-module). We also present functions to construct central extensions of \(U\) by \(G\).

Denote by \(Z^2(G,U)\) the abelian group of all cocycles from \(G\) to \(U\), under pointwise multiplication. The values \(\psi(g,h)\) of \(\psi \in Z^2(G, U)\) may be represented as a “cocyclic matrix” with entries in \(U\).

If \(\phi \from G \ar U\) is a set map with \(\phi (1_G) = 1_U\), then there is a coboundary \(\partial \phi \in Z^2(G,U)\) defined by \(\partial \phi(g,h) = \phi(g) \phi (h) \phi(gh)^{-1}\). The group of all coboundaries from \(G\) to \(U\) is denoted \(B^2(G,U)\), and we have \(H^2(G,U) = Z^2(G,U)/B^2(G,U)\). Then \(H^2(G,U)=I\times T\), where \(I\) is the (faithful) image of \(\operatorname{Ext}(G/G', U) \leq H^2(G/G', U)\) under inflation, and \(T\) is the (faithful) image of \(\Hom(H_2(G), U)\) under a certain transgression homomorphism. Here we provide functions which construct representatives for the elements in a generating set for each of these two factors.

For details of the theory and the algorithm used, see [FO00].

SetVerbose ("Cocycle", 1) will provide additional information on the calculations in the functions.

ExtGenerators(G, U): GrpPC, GrpPC -> [<AlgMatElt, RngIntElt>]: Given a soluble group \(G\) and an abelian group \(U\) (both defined by pc-presentations) the function returns a sequence of tuples describing generators for \(\operatorname{Ext}(G/G', U)\) as cocyclic matrices; the first entry in each tuple is a representative of a generator, the second is the order of the coset of the representative in \(H^2(G, U)\).

HomGenerators(G, U): GrpPC, GrpPC -> [<AlgMatElt, RngIntElt>]: Given a soluble group \(G\) and an abelian group \(U\) (both defined by pc-presentations) the function returns a sequence of tuples describing generators for \(\Hom(H_2 (G), U)\) as cocyclic matrices; the first entry in each tuple is a representative of a generator, the second is the order of the coset of the representative in \(H^2(G, U)\).

ElementSequence(G): GrpPC -> SeqEnum: For a soluble group \(G\), the function returns an indexed set of elements of \(G\) listed in the order used by ExtGenerators and HomGenerators.

RepresentativeCocycles(G, U, Ext, Hom): GrpPC, GrpPC, [AlgMatElt], [AlgMatElt] -> [AlgMatElt]: Let \(G\) be a soluble group \(G\) and \(U\) be an abelian group both defined by pc-presentations. Let \(Ext\) and \(Hom\) be the values returned by calling ExtGenerators and HomGenerators respectively. The function RepresentativeCocycles returns a complete and irredundant set of representatives for the elements of \(H^2(G,U)\) as cocyclic matrices.

CentralExtension(G, U, A): GrpPC, GrpPC, AlgMatElt -> GrpPC: Let \(G\) be a soluble group \(G\) and \(U\) be an abelian group, both defined by pc-presentations. Further, let \(A\) be a cocyclic matrix (as determined by the function RepresentativeCocycles). Then, this function returns the central extension of \(U\) by \(G\) determined by the cocyclic matrix \(A\).

CentralExtensions(G, U, Q): GrpPC, GrpPC, [AlgMatElt] -> [GrpPC]: If \(G\) is a soluble group \(G\) and \(U\) is an abelian group, both defined by pc-presentations, and \(Q\) is a sequence of cocyclic matrices (as determined by the function RepresentativeCocycles), this function returns the corresponding sequence of central extension of \(U\) by \(G\) determined by the sequence of cocyclic matrices \(A\). Note that the central extensions thereby constructed need not be mutually non-isomorphic.

CentralExtensionProcess(G, U): GrpPC, GrpPC -> Proc: Given a soluble group \(G\) and an abelian group \(U\) (both defined by pc-presentations) the function creates a process \(P\) for central extensions of \(U\) by \(G\). Note that the list of central extensions constructed by this process will contain all isomorphism types but the extensions need not be mutually non-isomorphic.

NextExtension(~P): Rec -> GrpPC: Given a central extension process \(P\), construct the next central extension determined by \(P\).

IsEmpty(P): Rec -> BoolElt: Return true if all central extensions determined by the process \(P\) have been constructed; otherwise return false.

Example: Central Extensions of a Dihedral Group (ex-b4d774)

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We compute the abelian invariants of \(H^2 (D_4, C_2)\).

> G := DihedralGroup(GrpPC, 4);
> U := AbelianGroup(GrpPC, [2]);
>
> Ext := ExtGenerators(G, U);
> Ext[1];
    <[Id(U) Id(U) Id(U) Id(U) Id(U) Id(U) Id(U) Id(U)]
    [Id(U) Id(U) Id(U) Id(U) Id(U) Id(U) Id(U) Id(U)]
    [Id(U) Id(U) Id(U) Id(U) Id(U) Id(U) Id(U) Id(U)]
    [Id(U) Id(U) Id(U) Id(U) Id(U) Id(U) Id(U) Id(U)]
    [Id(U) Id(U) Id(U) Id(U) U.1 U.1 U.1 U.1]
    [Id(U) Id(U) Id(U) Id(U) U.1 U.1 U.1 U.1]
    [Id(U) Id(U) Id(U) Id(U) U.1 U.1 U.1 U.1]
    [Id(U) Id(U) Id(U) Id(U) U.1 U.1 U.1 U.1], 2>,
>
> Hom := HomGenerators(G, U);
> Hom;
[
    <[Id(U) Id(U) Id(U) Id(U) Id(U) Id(U) Id(U) Id(U)]
    [Id(U) U.1 U.1 Id(U) Id(U) U.1 U.1 Id(U)]
    [Id(U) Id(U) Id(U) Id(U) Id(U) Id(U) Id(U) Id(U)]
    [Id(U) U.1 U.1 Id(U) Id(U) U.1 U.1 Id(U)]
    [Id(U) Id(U) Id(U) U.1 Id(U) U.1 U.1 U.1]
    [Id(U) U.1 Id(U) Id(U) U.1 U.1 Id(U) U.1]
    [Id(U) Id(U) Id(U) U.1 Id(U) U.1 U.1 U.1]
    [Id(U) U.1 Id(U) Id(U) U.1 U.1 Id(U) U.1], 2>
]
>
> AbelianInvariants(Ext, Hom);
[ 2, 2, 2 ]

We now compute the central extension of U by G determined by a single cocyclic matrix.

> A := RepresentativeCocycles(G, U, Ext, Hom);
> E := CentralExtension(G, U, A[2]);
> E;
GrpPC : E of order 16 = 2^4
PC-Relations:
    E.1^2 = E.4,
    E.2^2 = E.3 * E.4,
    E.2^E.1 = E.2 * E.3

Alternatively we can build all central extensions of U by G.

> E := CentralExtensions(G, U, A);
> "Number of extensions is ", #E;
Number of extensions is  8

Next, we provide an example of using the central extension process. Firstly, we create the groups and initialize the process.

> G := SmallGroup(12, 5);
> U := AbelianGroup(GrpPC, [2, 3]);
> P := CentralExtensionProcess(G, U);

Now we run over the central extensions and count conjugacy classes.

> C := [];
> while IsEmpty(P) eq false do
>    NextExtension(~P, ~E);
>    Append(~C, #Classes (E));
> end while;
> "# conjugacy classes is ", C;
# conjugacy classes is  [ 45, 72, 45, 72, 45, 72, 45, 72, 45, 72, 45,
72, 45, 72, 45, 72, 45, 72, 45, 72, 45, 72, 45, 72 ]

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Central Extensions#

Central Extensions