Let $G$ be a finite simple group and $M$ and $M'$ be two maximal subgroups of $G$. Also let $m_{M}$ be the set of minimal subgroups of $M$ and similarly $m_{M'}$ be the set of minimal subgroups of $M'$. Is it possible that $m_{M}\subseteq m_{M'}$.
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$\begingroup$ Can you explain what do you mean by minimal subgroup ? $\endgroup$– user21230Commented Jan 8, 2018 at 8:40
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Yes. Let $G = {\rm PSL}(3,4)$, and let $M$ be a maximal subgroup with the structure $3^2:Q_8$.
The minimal subgroups (i.e. the subgroups of prime order) of $M$ have order $2$ or $3$ and generate a subgroup of $M$ with structure $3^2:2$. This subgroup is also contained in a maximal subgroup $M'$ of $G$ with $M' \cong A_6$.
Another example is $G = M_{22}$ with $M = M_{10} = A_6 \cdot 2$ and $M' = 2^4:A_6$.
Yet another is $G = {\rm PSL}(3,7)$ again with $M = 3^2:Q_8$ and $M' = (3 \times A_4):2$.
It seems likely that there will be infinitely many examples of type $G = {\rm PSL}(3,q)$ and $M = 3^2:Q_8$, , and this should not be hard to prove, since the maximal subgroups of ${\rm PSL}(3,q)$ are well understood.
answered Jan 8, 2018 at 9:21
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$\begingroup$ I have found some more examples. I would conjecture that there are infinitely many. $\endgroup$ Commented Jan 8, 2018 at 10:45
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$\begingroup$ Thank you for explanation what minimal groups are. One idea which is coming to human mind is to check intersections of maximal groups of given finite simple group. Sometimes it is empty. I tested for $M_{22}$ and sizes of intersections of 8 maxes (sizes
[ 20160, 5760, 2520, 2520, 1920, 1344, 720, 660 ]) are:[ 360, 168, 168, 960, 96, 60, 60, 72, 60, 48, 24, 24, 12, 72, 24, 24, 6, 60, 24, 12, 6, 6, 16, 20, 12, 8, 12, 10 ].... and for $M_{23}$ are[ 1920, 5760, 1344, 660, 288, 11, 1920, 192, 60, 96, 1, 120, 48, 48, 1, 24, 48, 1, 48, 1, 1 ]$\endgroup$– user21230Commented Jan 9, 2018 at 10:04