Holomorphs

Given a group \(G\) and the full group of automorphisms \(A\) of \(G\) then the holomorph of \(G\) is the semidirect product \(G \times_\theta A\), where \(\theta: A \rightarrow Aut(G)\) is the identity map.

Holomorph(G): Grp -> GrpPerm, HomGrp, HomGrp

Holomorph(GrpFP, G): Grp -> Cat[GrpFP], GrpFP, HomGrp, HomGrp

Given a finite permutation, matrix, or pc-group \(G\) with full group of automorphisms \(A\), this function returns the semidirect product \(E\) of \(G\) by \(A\). The group \(E\) is returned as a permutation group (or a finitely presented group if GrpFP is specified) of degree \(|G|\) in which \(G\) is a regular normal subgroup, and \(A\) is the stabilizer of the point \(1\). The embedding map \(G \ar E\), and the natural epimorphism \(E \ar A\) are also returned. In the returned group \(E\), the generators of \(G\) appear first, followed by those of \(A\).

Holomorph(G, A): Grp, GrpAuto -> GrpPerm, HomGrp, HomGrp

Holomorph(GrpFP, G, A): Cat[GrpFP], Grp, GrpAuto -> GrpFP, HomGrp, HomGrp

Given a finite permutation, matrix or pc-group \(G\) and a group of automorphisms \(A\), this function returns the semidirect product \(E\) of \(G\) by \(A\). The group \(E\) is returned as a permutation group (or a finitely presented group if GrpFP is specified) of degree \(|G|\) in which \(G\) is a regular normal subgroup, and \(A\) is the stabilizer of the point \(1\). The embedding map \(G \rightarrow E\), and the natural epimorphism \(E \rightarrow A\) are also returned. In the returned group \(E\), the generators of \(G\) appear first, followed by those of \(A\).

Example: Holomorph of PGL(2, 9) (ex-fba5ab)

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We construct the holomorph of the group \(G = PGL(2, 9)\).

> G := PGL(2, 9);
> E := Holomorph(G); E;
Permutation group E acting on a set of cardinality 720
> #E;
1036800
> CompositionFactors(E);
    G
    |  Cyclic(2)
    *
    |  Cyclic(2)
    *
    |  Cyclic(2)
    *
    |  Alternating(6)
    *
    |  Alternating(6)
    1

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