When constructing examples of nonabelian finite $p$-groups with abelian automorphism group (and certain other desired properties), the authors of papers like http://arxiv.org/pdf/1304.1974v1.pdf leave the checks of certain basic properties of the group (like its order, its exponent,...) to the reader. These checks are easy when one can use an appropriate normal form theorem for the presentation, which, in each single case, one can guess and then show with the standard van der Waerden trick by considering an appropriate action of the group on the set of supposed normal forms.

My question is: Does one really need to guess a normal form theorem in each single case or is there some general theorem allowing for a more systematic approach at least in the case of finite $p$-groups of nilpotency class $2$ (which such papers tacitly use)? I am thinking of something where the presentation is assumed to be given in a "nice" way such as in the paper linked above, and the theorem states that elements, written as products of powers of the generators with powers of commutators of the generators, which are not equal for "obvious" reasons are distinct.

Thanks in advance!

  • 4
    $\begingroup$ Any nilpotent group has a normal form consisting of products of powers of generators coming from a power-commutator presentation (and this generalizes to power-conjugate presentations of polycyclic groups). Beyond that, I am not clear what you are asking for. $\endgroup$
    – Derek Holt
    Sep 13, 2014 at 9:31
  • $\begingroup$ @DerekHolt: Thank you! This is precisely what I was looking for. $\endgroup$ Sep 13, 2014 at 11:26


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