Dtyjlui

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $alpha$ of $G$ is called a class-preserving if for each $xin G$, there exists an element $g_xin G$ such that $alpha(x)=g_x^{-1}xg_x$. An automorphism $alpha$ of $G$ is called a $2$nd class-preserving if for each $xin G$, there exists an element $g_xin G'=[G,G]$ such that $alpha(x)=g_x^{-1}xg_x$. Let $mathrm{Aut_c}(G)$ and $mathrm{Aut_c^2}(G)$ respectively denote the group of all class-preserving and $2$nd class-preserving automorphisms of $G$.

I have made a GAP program to find the structure of $mathrm{Aut_c}(G)$ but I failed to make a GAP program to find the structure of $mathrm{Aut_c^2}(G)$. The GAP program to find the structure of $mathrm{Aut_c}(G)$ is following:

ClassPreservingAuts:= function(G)

local A,I,cc,gens,auts,a,ok,i,hom;

A:=AutomorphismGroup(G);

I:=InnerAutomorphismsAutomorphismGroup(A);

hom:=NaturalHomomorphismByNormalSubgroup(A,I);

cc:=ConjugacyClasses(G);

gens:=;

auts:=Group([One(A)]);



# check for class preserving

for a in Elements(A) do

  ok:=true;

  # run through classes

  i:=0;

  while i<Length(cc) and ok=true do

    i:=i+1;

    if not (Representative(cc[i])^a in cc[i]) then

      ok:=false;

    fi;

  od;

  # a is class preserving

  if ok=true and not (a in auts) then

    Add (gens,a);

    auts:= Group(gens);

    #inng:=Image(hom(x));

    #gens:=GeneratorsOfGroup(inng);

  fi;

od;

return auts;

return auts/I;

return Size(auts)/Size(I);

end;

My question is the following:

Can anybody help me to make a GAP program to find the structure of $mathrm{Aut_c^2}(G)$?

Your current test is

if not (Representative(cc[i])^a in cc[i]) then

that will eliminate elements as not lying in the subgroup you want. We could phrase this alternatively (this is in fact what the in test does) as

if RepresentativeAction(G,Representative(cc[i])^a,Representative(cc[i])=fail then

that is we are testing whether there is an element in G that will conjugate the class representative in the same way as the automorphism a does.

This is now easily generalized. Add

Gd:=DerivedSubgroup(G);

at the start and change the test to

if RepresentativeAction(Gd,Representative(cc[i])^a,Representative(cc[i])=fail then

Finally -- it is not clear from your code whether you want to factor out the inner automorphisms -- you would factor out not all inner automorphisms, but only those induced by $G'$.