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I am revising for my Group Theory exam and am stuck on the following question;

The Frattini subgroup $\Phi(G)$ of a group $G$ is defined to be the intersection of all maximal subgroups of $G$. Prove that $\Phi(G)$ is a characteristic subgroup of G.

Why is this the case?

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2 Answers 2

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HINT. If $H$ is a maximal subgroup of $G$, and $f\colon G\to K$ is a homomorphism with kernel contained in $H$, then $f(H)$ is a maximal subgroup of $f(G)$.

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let $\alpha$ be an arbitrary automorphism of $G$. If $M$ be a maximal subgroup of $G$, then $\alpha(M)$ is a maximal subgroup, too. then $\alpha(\phi(G))=\alpha(\cap M)=\cap \alpha(M)=\cap M'=\phi(G)$, M' is a maximal subgroup.

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