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 = Cp2, where p is prime, generated by a, say; here, Φ(G) = < ap >.

See also: