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

90

ABELIAN GROUPS

[CH. IV

elements of G whose orders divide p2 constitute a group of
order p2a-', where 3 is the number of invariants of G which
are equal to p. In general, let xi, x2,..., xx represent the
number of the invariants of G which are equal to p, p2,... pX
respectively, and suppose that they include all the invariants
of G. The number N of the elements of G whose orders divide
pk, k < X, is then given by the formula
Nypxl+22+... +(k-l)Xl+k(xk+x +l.. +XX).
It should be observed that the orders of the independent
cyclic groups which generate a given group G are completely
determined by G when G is of order p'. In general, G is the
direct product of Sylow subgroups and hence it is also the
direct product of a series of cyclic independent subgroups
C1, C2,..., Cm', each of which has for its order a power of a
prime number. Unless the contrary is stated it will be assumed
that the order of each of these subgroups exceeds unity, and
hence their number and their orders are completely determined
by G; and, in turn, they determine G completely. That is,
if these orders are the same for two groups these groups are
simply isomorphic. These orders are therefore invariants
of G, but they are not the only numbers which are known as
invariants of G. Their number constitutes the largest possible
number of orders of independent cyclic groups in G; that is,
neither G nor any of its subgroups can have more than m'
independent generators.
It is important to note that the term set of independent generators as regards abelian groups is usually employed to represent a set of independent generators which is such that the
group generated by an arbitrary number of them has only the
identity in common with the group generated by the remaining
ones. In dealing with abelian groups we shall always use this
term with this special meaning and not with its general meaning
given in ~ 3.
36. Largest and Smallest Number of Possible Invariants.
We proceed to find the smallest number of independent generators of G. The given subgroups C1, C2,..., Cm' can be