Timeline for General bound for the number of subgroups of a finite group

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May 3, 2019 at 11:54	comment	added	Geoff Robinson		I think that if $G$ is a direct product of cyclic groups of distinct prime orders, every subgroup of $G$ is cyclic, so this seems be different from that suggested by your second sentence when there are three or more different primes (and $G$ is cyclic of squarefree order). Also, by a theorem of Guralnick and Lucchini (which uses classification of finite simple groups), that if $\|G\| = \prod_{i=1}^{m}p_{i}^{a_{i}}$ with the $p_{i}$ distinct primes, and $a_{j} = {\rm max}\{a_{i}\}$, then each subgroup of $G$ is generated by at most $a_{j}+1$ elements, usually less that $\Omega(n)-1$
May 3, 2019 at 10:11	history	answered	Adam P. Goucher	CC BY-SA 4.0