# Frattini subgroup

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In mathematics, the **Frattini subgroup** Φ(*G*) of a group *G* is the intersection of all maximal subgroups of *G*. (If *G* has no maximal subgroups, then Φ(*G*) is defined to be *G* itself.)

## Some facts

- Φ(
*G*) is equal to the set of all**non-generators**or**non-generating elements**of*G*. A non-generating element of*G*is an element that can always be removed from a generating set; that is, an element*a*of*G*such that whenever*X*is a generating set of*G*,*X*− {*a*} is also a generating set of*G*. - Φ(
*G*) is always a characteristic subgroup of*G*; in particular, it is always a normal subgroup of*G*. - If
*G*is finite, then Φ(*G*) is nilpotent.

An example of a group with nontrivial Frattini subgroup is the cyclic group *G* = *C*_{p2}, where *p* is prime, generated by *a*, say; here, Φ(*G*) = < *a*^{p} >.

See also: