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This editable Main Article is under development and subject to a disclaimer.

[edit intro]

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

Formally,

Φ ( G ) = G ∩ ⋂ M M {\displaystyle \Phi (G)=G\cap \bigcap _{M}M\,}

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