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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 ?

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    $\begingroup$ It depends on the factorization of $2^p-1$, see groupprops.subwiki.org/wiki/… $\endgroup$ – Glorfindel Jul 8 '17 at 12:39
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    $\begingroup$ 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. $\endgroup$ – Glorfindel Jul 8 '17 at 12:51
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    $\begingroup$ @PéterKomjáth: No, we have then $2^p \equiv 2$ (mod $q^3$), respectively, $2^{p-1} \equiv 1$ (mod $q^3$). $\endgroup$ – Stefan Kohl Jul 8 '17 at 13:22
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    $\begingroup$ I did a quick check for $p$ at most $800$ or so and found no examples. In the end this is a question about number theory and not group theory, and it might be difficult. For example it is an old open problem to determine whether $2^p - 1$ is squarefree for all primes $p$. If the answer to that old problem is yes, then you are asking whether every group of order $2^p - 1$ is cyclic. $\endgroup$ – spin Jul 8 '17 at 18:59
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    $\begingroup$ msp.org/pjm/1967/22-3/pjm-v22-n3-p15-p.pdf $\endgroup$ – Mark Sapir Jul 8 '17 at 19:26

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