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

~ 48] INTRODUCTION TO PRIME-POWER GROUPS

119

theorem was first proved by L. Sylow * and it may be regarded
as the most important theorem relating to the prime-power
groups.
It has been observed that the totality of the invariant
operators of any non-abelian group constitutes an important
subgroup known as the central. With respect to its central,
G is isomorphic to a group of order pm', m' <m. This quotient
group must also have a central subgroup, if it is non-abelian,
and this gives rise to a second quotient group of order pm",
m"<m'. By continuing this process we must arrive at an
abelian quotient group. It is a matter of considerable importance to observe that this abelian group is never cyclic. In
fact, this is a special case of the theorem proved in ~ 28 that
the central quotient group of a non-abelian group is always
non-cyclic.
The subgroup of G which corresponds to an invariant subgroup of order p in the central quotient group of G is abelian,
but includes operators which are not in the central of G. Hence
it results that every non-abelian group of order pm contains an
invariant abelian subgroup whose operators are not separately
invariant under the group.
Since we can always arrive at the identity by forming
successive central quotient groups of G it results that G must
have at least one invariant subgroup whose order is an arbitrary divisor of the order of G. Suppose that H1, H2,..
Hp represent any complete set of conjugate subgroups of G.
Since each of these subgroups is transformed into itself by a
subgroup of G whose order is a power of p, it results that p
is also a power of p. Hence each of these H's must transform
into itself each one of at least p-1 of the other H's, since
it transforms itself into itself, and since it must transform a
multiple of p of these conjugates among themselves. That
is, each one of a complete set of conjugate subgroups of a group of
order pm is transformed into itself by at least p- 1 others of the set,
if the set includes more than one subgroup. In particular, every
subgroup of order p"-1 in a group of order pm is invariant under
* Mathematische Annalen, vol. 5 (1872), p. 584.