1
$\begingroup$

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 $x\in G$, there exists an element $g_x\in G$ such that $\alpha(x)=g_x^{-1}xg_x$. An automorphism $\alpha$ of $G$ is called a $2$nd class-preserving if for each $x\in G$, there exists an element $g_x\in G'=[G,G]$ 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)]);

$\sharp$ check for class preserving

for a in Elements(A) do

ok:=true;

$\sharp$ 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;

$\sharp$ a is class preserving

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

Add (gens,a);

auts:= Group(gens);

$\sharp$inng:=Image(hom(x));

$\sharp$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 $\mathrm{Aut_c^2}(G)$?

$\endgroup$
  • 4
    $\begingroup$ Cross posted to math.stackexchange.com/questions/3076661 There is a GAP forum mailing list, and you could ask there. $\endgroup$ – Derek Holt Jan 17 at 7:43
  • $\begingroup$ Couldn't you adapt your program for Aut_c(G) by only including those automorphisms where g_x is in the commutator? $\endgroup$ – tj_ Jan 18 at 10:27
  • 1
    $\begingroup$ I'm voting to close this question as off-topic because this is better suited for a GAP forum as Derek Holt suggests. $\endgroup$ – Neil Hoffman Jan 18 at 20:21
  • $\begingroup$ First note that for a subgroup $U \le G$ the relation $g \sim h :\Leftrightarrow \exists u \in U: g=uhu^{-1}$ defines an equivalence relation on $G$ (I don't know if it has a name). In your case take $U:=G'$ the commutator. Then the following should work: (1) Replace cc:=ConjugacyClasses(G) by a set of $R$ of representatives for $\sim$ (2) Replace Representative(cc[i])^a in cc[i]) by a check if $a(g)$ and $g$ are in the same equivalence class for $g \in R$. -- These tasks are supported in GAP by the functions EquivalenceClasses and EquivalenceClassOfElement. $\endgroup$ – tj_ Jan 25 at 7:13

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.