# Frattini subgroup

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

Formally,

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

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

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

## References

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