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

88

ABELIAN GROUPS

[CH. IV

element of (si). That is, the pa2th power of every element
of G is in (si) but the pa2-i power of s2' is not in (si). We
proceed to prove that s2' may be so selected that the cyclic
groups (si) and (s'2) have only the identity in common; that is,
S2P2 =1
Since s2/pa2 is in (si), and the order of s2' does not exceed
that of Si, there must be an element in (si) whose pa2th power
is the inverse of s2'Pa2. The product of s2' and this element is
therefore of order pa2. Moreover, the p"a2-th power of this
product is not contained in (si) since one of its factors is in
(Si), while the other does not have this property. In what
follows we shall denote this product, of order pa2, by s2. If
the order of G is the product of the orders of Si and s2 it is clear
that G is the direct product of (si) and (s2). We proceed to
prove that G is always the direct product of cyclic groups.*
The orders of these cyclic groups are called the invariants t
of G. In particular, if Si is of order pal and if G is the direct
product of (si), (s2), the invariants of G are pal, pa2.
If G contains elements which are not included in the group
(si, s2), generated by sl and s2, we may suppose that the p"ath
power of every other element is in (si, s2) while the p"a-1th
powers of some of the elements are not in this subgroup. We
proceed to prove that at least one of the latter elements is
of order pa. Let S3' be any one of these elements. As
Sta3
is in (sl, S2), and as 3 a2 <oa1, there must be some element in (Sl, s2) whose p"3th power is the inverse of s'3.
The product, S3, of this element and s'3 is therefore of order pa,
and G involves the direct product of the three groups (sl),
(S2), (sa). As this process may clearly be continued until all
* It is implied that each of these cyclic groups has only the identity in common
with the group generated by all the others. In other words, they are independent
cyclic groups.
t The invariants of an abelian group have also been called the elementary
divisors of the order of this group. Frobenius and Stickelberger, Crelle, vol.
86 (1878), p. 238.