Delta subgroup with GAP

Question

This question is perhaps a bit silly, but I haven't figured out from the GAP manual how to do this. I want to create a function delta which takes as input a (finite) group $G$ and returns the full preimage of $Z(G/\Phi(G))$, where $\Phi(G)$ is the Frattini subgroup of $G$.

Any help would be appreciated.

By $Z (G/\Phi (G))$ is meant the center of the quotient group, and by full preimage is meant the subgroup of $G$ which maps to that center under the quotient map? — hardmath, Dec 11 '17 at 16:04
Yes. So $\Delta(G)/\Phi(G) = Z(G/\Phi(G))$ and I want to find $\Delta(G)$. — the_fox, Dec 11 '17 at 16:05

score 3 · Accepted Answer · answered Dec 11 '17 at 20:29

You need to form a natural homomorphism to go between $G$ and a factor group. Then the function is simply:

DeltaSubgroup:=function(G)
local hom;
  hom:=NaturalHomomorphismByNormalSubgroup(G,FrattiniSubgroup(G));
  return PreImage(hom,Center(Image(hom,G)));
end;

Delta subgroup with GAP

1 Answers1