70.4 Automorphisms

Automorphisms( P )
OuterAutomorphisms( P )
NormalizedAutomorphisms( P )
NormalizedOuterAutomorphisms( P )

all return a record with components

sizeOutG:

the size of the group of outer automorphisms of P,

sizeInnG:

the size of the group of inner automorphisms of P,

sizeAutG:

the size of the full automorphism group of P,

generators:

a list of group automorphisms that generate the group of all, outer, normalized or normalized outer automorphisms of the polycyclically presented p-group P, respectively. In the case of outer or normalized outer automorphisms, this list consists of preimages in Aut(P) of a generating set for Aut(P)/Inn(P) or Aut_n(P)/Inn(P), respectively.

    gap> g:= SolvableGroup( "Q8" );;
    gap> Automorphisms( g );
    rec(
    sizeAutG := 24,
    sizeInnG := 4,
    sizeOutG := 6,
    generators := 
    [ GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ b, a, c ] ), 
      GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a*b, b, c ] ), 
      GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a, b*c, c ] ), 
      GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a*c, b, c ] ) ] )
    gap> OuterAutomorphisms( g );
    rec(
    sizeAutG := 24,
    sizeInnG := 4,
    sizeOutG := 6,
    generators := 
    [ GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ b, a, c ] ), 
      GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a*b, b, c ] ) ] ) 

Note: If the component P.isCompatiblePCentralSeries is not bound it is computed using IsCompatiblePCentralSeries.

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