[GAP Forum] Testing action of automorphisms on groups

[Stephen Linton]

To apply an automorphism to an element, you use Image(a1, x1), not a1(x1).

	Steve

On 23/02/2020, 11:10, "Siddhartha Sarkar" <siddhartha18 at gmail.com> wrote:

    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
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