Frattini subgroup

Some facts

• is equal to the set of all non-generators or non-generating elements of. A non-generating element of is an element that can always be removed from a generating set; that is, an element a of such that whenever is a generating set of containing a, is also a generating set of.
• is always a characteristic subgroup of ; in particular, it is always a normal subgroup of.
• If is finite, then is nilpotent.
• If is a finite p-group, then. Thus the Frattini subgroup is the smallest normal subgroup N such that the quotient group is an elementary abelian group, i.e., isomorphic to a direct sum of cyclic groups of order p. Moreover, if the quotient group has order p^k, then k is the smallest number of generators for . In particular a finite p-group is cyclic if and only if its Frattini quotient is cyclic. A finite p-group is elementary abelian if and only if its Frattini subgroup is the trivial group,
• If and are finite, then.