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 ![]() |
[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) |
Frattini subgroup. Marcel Herzog (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frattini_subgroup&oldid=14968