[GAP Forum] Testing action of automorphisms on groups

[Siddhartha Sarkar]

Dear forum,

I am trying to compute among the class 2 groups of order 2^7 which ones
admit
a non-inner automorphism of order 2 that fixes the Frattini subgroup
elementwise. I tried to check it manually before writing a code:

gap> LoadPackage("autpgrp");

true

gap> all:=AllSmallGroups(2^7);;

gap> lst:=Filtered(all,g->NilpotencyClassOfGroup(g)=2);; Length(lst); 947

gap> g:=lst[1];

<pc group of size 128 with 7 generators>

gap> AutomorphismGroup(g);

<group of size 6144 with 12 generators>

gap> A := AutomorphismGroupPGroup(g);;

gap> B:=PcGroupAutPGroup(A);

<pc group of size 6144 with 12 generators>

gap> I:=InnerAutGroupPGroup(B);

Group([ f4, f3, <identity> of ..., <identity> of ..., <identity> of ...,

  <identity> of ..., <identity> of ... ])

gap> elmsg:=Elements(g);;

gap> elmsaut:=Elements(B);;

gap> elmsinn:=Elements(I);;

gap> elmsaut0:=Difference(elmsaut,elmsinn);;

gap> elmsaut02:=Filtered(elmsaut0,y->Order(y)=2);;

gap> Fratg:=FrattiniSubgroup(g);

Group([ f3, f4, f5, f6, f7 ])

gap> elmsFratg:=Elements(Fratg);;

gap> a1:=elmsaut0[1];x1:=elmsFratg[1];

f1

<identity> of ...

gap> a1(x1);

Error, no method found! For debugging hints type ?Recovery from
NoMethodFound

Error, no 1st choice method found for `CallFuncList' on 2 arguments at
/Users/sidhu/Desktop/GAP/gap-4.10.2/lib/methsel2.g:250 called from

<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )

 called from read-eval loop at *stdin*:21

type 'quit;' to quit to outer loop

brk> quit;

gap> KnownAttributesOfObject(a1);

[  ]

gap> KnownAttributesOfObject(x1);

[  ]


i.e., GAP store the elements of automorphisms as plain list without knowing
what it does to the elements of the group.

What is a better option?

Best regards,

Siddhartha