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

120

PRIME-POWER GROUPS

[CH. V

this group. As has been observed in ~ 28, this theorem is also
a special case of the following theorem: If Hi and H2 are two
conjugate subgroups of G, then the index of H1 or H2 under G
is greater than the index under H1 or H2 of the cross-cut of H1
and H2.
49. Invariant Abelian Subgroups. From the fact that every
group of order pm contains at least p invariant operators and
that its central quotient group has the same property, it results
that every group of order pm, m> 2, contains a subgroup of
order p"-l which involves p2 invariant operators. In a similar
way we observe that every group of order pm, m> 5, contains
a subgroup of order pm-1-2 which involves p3 invariant operators. In general, every group of order pm, m> (a+2)(a- 1)/2,
contains a subgroup of order
pem-l-2-3-... -(a-l) = pm-c(a-l)/2
which involves pa invariant operators. As the group formed
by these invariant operators corresponds to an invariant subgroup in the quotient group, it results that it is invariant under
the entire group. Since every invariant subgroup of order pa
in a group of order pm is contained in an invariant subgroup of
order p"+l, it results from the above that every group of order
p ", m>a(a+l)/2, contains an invariant abelian group of order
pa+l. In other words, every group of order pm, m>(3 —l)/2
contains an invariant abelian subgroup of order pi.
In the special case when p = 2 this theorem may be expressed
in a little more general form as follows:
Every group of order 2m, m > /(3 —1)/2, 0> 3, contains an abelian subgroup of order 2 *. The proof of this extended theorem
is short if the preceding developments are employed. In
fact, it has been proved that G involves a subgroup of order
pm —2-.. -(3) 2)-3) pm- (8-2)(#-3)/2 > 3,
* In fact, there is always an invariant abelian subgroup of order 20 when the
given conditions are satisfied. Cf. Messenger of Mathematics, vol. 41 (1912),
p. 28.