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# 6 Investigating the automorphism group

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.

- `AutomorphismGroupSequence(` PcpGroup`) F`

  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

  a[i][j] denotes the image of Out(G/gammaj+1G) in Out(G/gammai+1G).

  vskip3ex

  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 ] ]

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