Frattini Subgroup

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In group theory, the Frattini subgroup is the intersection of all maximal subgroups of a group.

Formally,

${\displaystyle \mathrm{\Phi }(G)=G\cap \bigcap _{M}M}$

where M runs over all maximal subgroups of G. If G has no maximal subgroups then ${\displaystyle \mathrm{\Phi }(G)=G}$.

The Frattini is a subgroup, which is normal and indeed characteristic.

References[edit]

• Marshall Hall jr (1959). The theory of groups. New York: Macmillan, 156.