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

92

ABELIAN GROUPS

ICrH. IV

priate to call the orders of these independent generators the
invariants of G, although the latter method has some advantages. The choice of invariants such that their number lies
between a and n' seems less natural. We evidently arrive
at the a invariants if we choose the independent generators
in the following way. Start with an element of highest order
and then select any other element such that the two generate
the largest possible subgroup. The orders of two independent
generators of this subgroup are the first two invariants of G.
If we add to this subgroup another element so that the three
generate the largest possible subgroup, we arrive at the third
invariant, etc.
A marked difference between the two given methods of
arriving at the invariants of G should perhaps be emphasized;
viz., the orders of the independent generators of G are completely
determined by mn', but not by a. That is, if two sets of m'
independent generating cyclic subgroups of G were given, the
orders of the subgroups of one set would be the same as those
of the other; but if two sets of a independent generating cyclic
subgroups of G were given, the orders of those of one set could
generally vary a great deal from the orders of those of the
other. A necessary and sufficient condition that the orders of
these two sets must be the same is that the a invariants of each
of the Sylow subgroups of G, with one possible exception, are
equal. The number a is said to be the rank of G.
If G is the direct product of a series of subgroups G1, G2,..Gx, we may select a set of independent generators of G
by combining arbitrary sets of independent generators of each
of these subgroups. Suppose that G1, G2,..., Gx are the
Sylow subgroups of G. Any element t of G will have a constituent, which may be the identity, from each one of these subgroups, and the order of t will be the product of the orders of
these constituents. To determine the number of the elements
of a given order in G it is only necessary to determine the number of elements of a given order in each of the Sylow subgroups.
That is, if the order of t is plalp2a2... pa (pi, p2,., pX
being prime numbers), the number of elements of G whose