Frattini-subgroup(2)

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The characteristic subgroup $\Phi(G)$ of a group $G$ defined as the intersection of all maximal subgroups of $G$, if there are any; otherwise $G$ is its own Frattini subgroup. It was introduced by G. Frattini [1]. The Frattini subgroup consists of precisely those elements of $G$ that can be removed from any generating system of the group containing them, that is, $$ \Phi(G) = \{ x \in G : \langle M,x \rangle = G \Rightarrow \langle M \rangle = G \} \ . $$

A finite group is nilpotent if and only if its derived group is contained in its Frattini subgroup. For every finite group and every polycyclic group $G$, the group $\Phi(G)$ is nilpotent.

References

[1]	G. Frattini, "Intorno alla generazione dei gruppi di operazioni" Atti Accad. Lincei, Rend. (IV) , 1 (1885) pp. 281–285
[2]	A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)