# $G$ a group, with $p$ a prime number, and $|G|=2^p-1$, is it abelian?

During my research I came across this question, I proposed it in the chat, but nobody could find a counterexample, so I allow myself to ask you :

$$G$$ a group, with $$p$$ a prime number, and $$|G|=2^p-1$$, is it abelian ?

• It depends on the factorization of $2^p-1$, see groupprops.subwiki.org/wiki/… Jul 8 '17 at 12:39
• Yes, but probably easier to solve. I doubt there is a prime $q$ for which $q^3 | 2^p - 1$, but the other 'case', two primes $q_1, q_2$ in the factorization for which $q_1 | q_2 - 1$ might very well be possible. Jul 8 '17 at 12:51
• @PéterKomjáth: No, we have then $2^p \equiv 2$ (mod $q^3$), respectively, $2^{p-1} \equiv 1$ (mod $q^3$). Jul 8 '17 at 13:22
• I have rolled back the edits to this old question, which seem to be purely cosmetic and done for promotion. Yes the question is appropriate for MO, but that doesn't mean you need to add some kind of "RL" label May 18 at 11:33
• If you are claiming that you can answer this question, then you can either add an answer below, or write up your work as a preprint in the normal academic way, or write it in a blog post and leave a link. Saying "I can answer this, can you?" is not a constructive use of this site, nor is it collegial. May 18 at 11:35