Is there a way to check if a group extension is split or not using GAP?

Question

I use the GAP command 'NormalSubgroups' to obtain the list of normal subgroups of a given group. Then I use 'FactorGroup' to construct the quotient of the given group by a normal subgroup. An example is given below.

G:=AlternatingGroup(4);; list:=NormalSubgroups(G);; N:=list[2];; H:=FactorGroup(G,N);;

Here, G is an extension of H by N. Is there a way to check if this extension is split using GAP commands?

Answer 1

The function ComplementClassesRepresentatives(G,N) finds representatives of the $G$-conjugacy classes of complements to $N$. It returns an empty list if the extension is not split. For example (in your case and a nonsplit case:

gap> ComplementClassesRepresentatives(G,N);
[ Group([ (2,4,3) ]) ]
gap> G:=SL(2,5);N:=Center(G);
SL(2,5)
<group of 2x2 matrices over GF(5)>
gap> ComplementClassesRepresentatives(G,N);
[  ]

The current methods available require that $N$ is solvable (calculated using 1-cohomology, respectively "Fox derivatives"), or that $G/N$ is solvable (using cyclic factors in a composition series) -- these seem to be the cases relevant in applications.