Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

72

ABSTRACT GROUPS

[CH. III

of G in such a manner that it includes any arbitrary one of the
operators of G which are not contained in H, while the remaining operators of the set belong to H. Moreover, there is at
least one maximal subgroup of G which does not include any
given one of the independent generators of a particular set of
independent generators of G. Hence it results that the ~-subgroup of G is the cross-cut of all the maximal subgroups of G. This
useful second definition of the 0-subgroup is also due to Frattini.
If a ~-subgroup of the group G involves a non-invariant
subgroup or a non-invariant operator, this subgroup or operator
cannot be transformed into all its conjugates under G by the
operators of the 0-subgroup. That is, every complete set of
conjugates of the ~-subgroup is an incomplete set of conjugates under G whenever the former set involves more than one
element. If this were not the case all the operators of G which
would transform one of these elements into itself would form
a subgroup which would not involve all the operators of the
~-subgroup of G. This subgroup could not be maximal, since
it does not involve the 0-subgroup. As any maximal subgroup
obtained by extending this subgroup by means of operators
of G could also not involve the 0-subgroup, we have proved
the theorem: If the ~-subgroup of a group G involves a noninvariant operator or subgroup, the number of conjugates under
G of this operator or subgroup is greater than the number of the
corresponding conjugates under the ~-subgroup.
An important special case of this theorem was noted by
Frattini, who observed that the 0-subgroup of any group involves
only one Sylow subgroup for every prime which divides the order
of the 0-subgroup. In other words, every 0-subgroup is the
direct product of its Sylow subgroups, and hence we can always
reach the identity by forming successive s-subgroups, starting
with any given group. If a group can be represented as a nonregular primitive substitution group of degree n, its n subgroups
of degree n-1, each being composed of all its substitutions
which omit a letter, are maximal and have only the identity
in common. Hence it results that the s-subgroup of every
primitive substitution group is the identity.