3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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
0
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.