How we can show that for group $G$ (finite non-abelian p-group, I don't know which ones are necessary) and $M$ maximal subgroup of $G$. We can have
$C_G(M)\le C_G(\Phi(G))\le Z(\Phi(G))$
$\Phi(G)$ denote frattini subgroup of G
Left relation kinda simple I don't know if it is true for infinite groups, I'm not sure how right relation is true
2026-03-25 10:55:28.1774436128
property about centralizer of maximal subgroup
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As mentioned in the comments, $C_G(M) \leq C_G(\Phi(G))$ follows since $\Phi(G) \leq M$.
Also, the second containment is false, for example it is possible that $\Phi(G)$ is central. Consider a nonabelian $p$-group of order $p^3$, or more generally any extraspecial $p$-group.