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

110

ABELIAN GROUPS

[Cu. IV

If G is an abelian group of order pm and of type (1, 1, 1,...),
it contains no characteristic subgroup besides the identity;
but every other abelian group contains at least one characteristic subgroup besides the identity. Suppose that G is an abelian
group of order pm but not of type (1, 1, 1,... ), and let pa
be one of its largest invariants. If exactly X1 of the invariants
of G are equal to pa', then G contains a characteristic subgroup
of order pX1 and of type (1, 1, 1,... ). This characteristic
subgroup C1 has been called the fundamental characteristic
subgroup * of G, since it is contained in every possible characteristic subgroup of G besides the identity, as we shall prove
in the following paragraph.
It is evident that the subgroup of order pX1, which is composed of the identity and of all the operators of order p which
are generated by the operators of highest order contained in G,
is the characteristic subgroup C1. Moreover, the conjugates
under the group of isomorphisms I of every operator of order p,
which is found in G but not in C1, generate a characteristic
subgroup of G which includes C1 as a subgroup. This fact
results immediately from the different possible ways of selecting the independent generators of G. If the second largest
invariants of G are pa2, and if there are exactly X2 such invariants in G, then G contains also a characteristic subgroup C2
of order pXl+X, which is composed of the identity and of the
operators of order p which are generated by the operators
of order pa2 contained in G. By continuing this process we
clearly arrive at the following
THEOREM. If a group G of order pm has Xi invariants which
are equal to pal, X2 which are equal to p"a,..., Xg which are
equal to pa", where ai>a2>... > a0, then G has 3 characteristic subgroups C1, C2,..., C0, besides the identity, such that
each of them is generated by operators of order p. Their orders
are px,l pXI+X2,..., pXl+X2+ *. +xp, respectively, and each is
included in all of those which follow it.
Since every operator of highest order contained in an abelian
group can be used as an independent generator of the group,
* American Journal of Mathematics, vol. 27 (1905), p. 15.