View source for Frattini-subgroup(2)

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The [[characteristic subgroup]] $\Phi(G)$ of a group $G$ defined as the intersection of all [[maximal subgroup]]s of $G$, if there are any; otherwise $G$ is its own Frattini subgroup. It was introduced by G. Frattini [[#References|[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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  G. Frattini,   "Intorno alla generazione dei gruppi di operazioni"  ''Atti Accad. Lincei, Rend. (IV)'' , '''1'''  (1885)  pp. 281–285</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Kurosh,   "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR>
</table>