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-generating elements of G; a non-generating element of G is an element that can always be removed from a generating set, i.e. it is an element a of G such that whenever X is a generating subset of G, X − {a} is also a generating subset 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:

• Fitting subgroup