# Thompson's Conjecture

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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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