Frattini subgroup

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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. 
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