Are there any results on the minimal number of generators required to give a presentation of a finite group? More specifically, given a group G, what is the minimal number of generators needed for a presentation of it? No bounds on the number of relations are assumed.

I've not found anything after doing some research.

  • $\begingroup$ Well, it depends on what the group is. Are there any specific cases you are interested in? $\endgroup$ – Eric Wofsey Oct 24 '18 at 2:07
  • $\begingroup$ This is the same as asking about the minimal number of generators for a group - presentations are not particularly relevant. This is a much studied question. For example it is known that all finite simple groups can be generated by two elements. $\endgroup$ – Derek Holt Oct 24 '18 at 7:55
  • $\begingroup$ The direct product of $n$ copies of the cyclic group of order $p$ prime, $\mathbb{Z}_p^n$, cannot be generated by fewer than $n$ elements (via a basic vector space argument). $\endgroup$ – user1729 Oct 24 '18 at 12:27

Well, there is the simple result that it can't possibly require more generators than the number of prime factors in the group order (counted with multiplicity).


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