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I have heard that the following is a conjecture due to Thompson: The number of maximal subgroups of a (finite) group $G$ does not exceed the order $|G|$ of the group.

My question is: did Thompson really conjecture this? If so, is there any literature on the subject?

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up vote 7 down vote accepted

I've seen this conjecture attributed to Wall (1961), for example: On a conjecture of G.E. Wall. This is a recent article (journal version appeared in 2007), and it gives a bunch of references. The conjecture remains open. Here is a very recent article which does not attack the conjecture itself, but uses it as an inspiration for a different conjecture.

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Wall proved it for solvable groups. Another proof (for solvable groups) is given in Pál Hegedűs, The number of maximal subgroups of a solvable group, Publ. Math. Debrecen 78 (2011), no. 3-4, 687–689, MR2867210. Another paper on the solvable case is Benjamin Newton, On the number of maximal subgroups of a finite solvable group, Arch. Math. (Basel) 96 (2011), no. 6, 501–506, MR2821467 (2012f:20092). – Gerry Myerson Jun 21 '12 at 0:35
Many thanks for the references. – the_fox Jun 21 '12 at 1:02
Question: if $P$ is a $p$-group and $M_1,M_2$ are maximal subgroups of $P$ such that $M_1/\Phi(P )=M_2/\Phi(P )$, does it follow that $M_1=M_2$? I need this in order to understand something Newton mentions in the first page of his paper. – the_fox Jun 21 '12 at 15:00
@Stefanos Yes, and this holds in greater generality: if $H$ is a normal subgroup of $G$, and $G_1,G_2$ are subgroups containing $H$, then $G_1/H=G_2/H$ if and only if $G_1=G_2$. Indeed, if there exists $a\in G_1\setminus G_2$, then the coset $aH$ is contained in $G_1/H$ but not in $G_2/H$. – user31373 Jun 21 '12 at 15:23
Thanks! My question was a bit silly, I have to admit! – the_fox Jun 21 '12 at 15:40

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