[GAP Forum] commutator relations in p-group

[Stefan Kohl]

Dear Forum,

Siddhartha Sarkar wrote:

> I need to know the classification of p-groups satisfy the following
> commutator relation:
> 
> (1) say the group G is minimally generated by d elements
>                                               x_1, x_2,...,x_d.
> (2) if d is even, the relation is [x_1, x_2]...[x_{d-1},x_d] = 1
> (3) for d odd, the relation is [x_1, x_2]...[x_d,x] = 1
>      for some element x in G.

Your condition is relatively weak --

A pc-presentation of a p-group of order p^n requires n(n+1)/2 relations.
Out of these you prescribe only one.

Given that there are asymptotically p^((2/27)n^3) isomorphism types
of groups of order p^n, it seems likely that there also is a huge
number of p-groups which satisfy your relation, and that it does not
make sense to attempt to classify them in general.

Bettina Eick has already suggested you off-list to use NQ or ANUPQ --
maybe these packages help you in investigating your problem.

By the way: when replying to Forum digests, please make sure that
you only cite the relevant part. Thanks!

Best wishes,

     Stefan Kohl