Frattini subgroup

Let denote an arbitrary group (finite or infinite) and let (respectively, ) mean that is a subgroup (respectively, a normal subgroup) of (cf. also Subgroup; Normal subgroup). Let denote a prime number.

The intersection of all (proper) maximal subgroups of is called the Frattini subgroup of and will be denoted by . If or is infinite, then may contain no maximal subgroups, in which case is defined as . Clearly, is a characteristic (hence normal) subgroup of (cf. also Characteristic subgroup).

The set of non-generators of consists of all satisfying the following property: If is a non-empty subset of and , then . In 1885, G. Frattini proved [a1] that is equal to the set of non-generators of . In particular, if is a finite group and for some subgroup of , then . Using this observation, Frattini proved that the Frattini subgroup of a finite group is nilpotent (cf. also Nilpotent group). This basic result gave its name. Moreover, his proof was very elegant: if denotes a Sylow -subgroup (cf. also Sylow subgroup; -group) of , then he proved that , whence, as remarked above, and the nilpotency of follows. Since then, the enormously useful result that if is a finite group, and is a Sylow -subgroup of , then , is usually referred to as the Frattini argument.

The Frattini subgroup of is strongly interrelated with the commutator subgroup . In 1953, W. Gaschütz proved [a2] that for every (possibly infinite) group one has , where denotes the centre of (cf. also Centre of a group). The stronger condition is equivalent to the property that all maximal subgroups of are normal in . It follows that if is a nilpotent group, then . If is a finite group, then, as discovered by H. Wielandt, is nilpotent if and only if . If is a finite -group, then , where is the subgroup of generated by all the -th powers of elements of . If is a finite -group, then . It follows that if is a finite -group, then is equal to the intersection of all normal subgroups of with elementary Abelian quotient groups .

The Frattini quotient has also some important properties. If is cyclic (cf. also Cyclic group), then is cyclic. If is finite, then is nilpotent if and only if is nilpotent. Moreover, if is finite and divides , then divides also . If is a finite -group, then is elementary Abelian and if is a -automorphism of which induces the identity on , then, by a theorem of Burnside, is the identity automorphism of .

Finally, let be a group of order and let . The Burnside basis theorem states that any minimal generating set of has the same cardinality , and by a theorem of Ph. Hall the order of divides , where .

General references for these and more specific results concerning the Frattini subgroup are [a3], [a4], [a5].

References

[a1]	G. Frattini, "Intorno alla generazione dei gruppi di operazioni" Rend. Atti Acad. Lincei , 1 : 4 (1885) pp. 281–285; 455–457
[a2]	W. Gaschütz, "Über die -Untergruppe endlicher Gruppen" Math. Z. , 58 (1953) pp. 160–170
[a3]	B. Huppert, "Endliche Gruppen" , I , Springer (1967)
[a4]	D.J.S. Robinson, "A course in the theory of groups" , Springer (1982)
[a5]	W.R. Scott, "Group theory" , Prentice-Hall (1964)