Frattini subgroup

Some facts

• \Phi(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 containing a, X \setminus {a} is also a generating set of G.
• \Phi(G) is always a characteristic subgroup of G; in particular, it is always a normal subgroup of G.
• If G is finite, then \Phi(G) is nilpotent.
• If G is a finite p-group, then \Phi(G)=G^p [G,G]. Thus the Frattini subgroup is the smallest (with respect to inclusion) normal subgroup N such that the quotient group G/N is an elementary abelian group, i.e., isomorphic to a direct sum of cyclic groups of order p. Moreover, if the quotient group G/\Phi(G) (also called the Frattini quotient of G) has order p^k, then k is the smallest number of generators for G (that is, the smallest cardinality of a generating set for G). In particular, a finite p-group is cyclic if and only if its Frattini quotient is cyclic (of order p). A finite p-group is elementary abelian if and only if its Frattini subgroup is the trivial group, \Phi(G)={e}.
• If H and K are finite, then \Phi(H \times K) = \Phi(H) \times \Phi(K).

An example of a group with nontrivial Frattini subgroup is the cyclic group G of order p^2, where p is prime, generated by a, say; here, \Phi(G) = \left\langle a^p\right\rangle.

References

• (See Chapter 10, especially Section 10.4.)

References

1. Frattini, Giovanni. (1885). "Intorno alla generazione dei gruppi di operazioni". Accademia dei Lincei, Rendiconti.