Let G be a finitely generated group. Then the term gammacG of the lower central series is fully invariant subgroup of G. Thus every automorphism alphainAut(G) induces an automorphism varphicinAut(G/gammacG). We obtain a homomorphism nuccolonAut(G)toAut(G/gammacG), alphamapstoalphac. This homomorphism map the inner automorphism Inn(G) onto Inn(G/gammacG) and thus we obtain a homomorphism
nuccolonOut(G)toOut(G/gammacG). Similar, for every dleqc, we obtain a homomorphism muc,dcolon Out(G/gammacG)toOut(G/gammadG). Since nud = nuccirc muc,d, this yields that
im(nud)leq...leqim(muc,d)leqim(muc-1,d) leq...leqim(mud,d) = Out(G/gammadG). This sequence can be used to guess the shape of im(nud) and therefore to guess the shape of Out(G)/kernud. The AutPGrp-Package can be used to compute the images im(muc,d) if the abelian quotient of G is elementary abelian. For further details we refer to EH09.
if the abelianization of PcpGroup is elementary abelian, this method computes a list of the images of the outer automorphism group of G/gammacG in Out(G/gammadG) for any dleqc with Out(G/gammadG) being still solvable. More precisely, the entry
In the following example we consider the nilpotent quotients of the Grigorchuk group and compute its outer automorphism group sequence.
gap> G := ExamplesOfLPresentations( 1 );; gap> A := AutomorphismGroupSequence( G, 5 );; [1,2]: ab [ [ 2, 1 ] ] [1,3]: ab [ [ 2, 1 ] ] [1,4]: ab [ [ 2, 1 ] ] [1,5]: ab [ [ 2, 1 ] ] [2,3]: ab [ [ 2, 1 ] ] [2,4]: ab [ [ 2, 1 ] ] [2,5]: ab [ [ 2, 1 ] ] [3,4]: id [ 16, 11 ] [3,5]: ab [ [ 2, 2 ] ] [4,5]: ab [ [ 2, 2 ] ]
[Up] [Previous] [Next] [Index]