Frattini subgroup

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

Formally,

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

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

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

References

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