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

~ 36]

NUMBER OF POSSIBLE INVARIANTS

91

arranged in rows such that the orders of all those in one row
are powers of the same prime, and such that the order of each
is equal to or greater than the order of the one which follows
it in the same row. In case the rows do not contain the same
number, the vacant places may be filled by the identity.
Arranging these rows in the form of a rectangle, we have
C1, C2,..., Ca
Ca+l, Ca+2, ~* * * C2a
By forming the products of all those in each column we
obtain a independent cyclic subgroups such that the order of
each is divisible by the orders of all those which follow it.
These subgroups form the smallest possible number of generating
cyclic subgroups of G. The orders of these subgroups are commonly called the invariants of G, since any other system of
independent generating subgroups in which the order of every
group is divisible by the order of every following group is
composed of groups which are simply isomorphic with these
products. It may be observed that a is the largest number of
invariants in a Sylow subgroup of G, while m' is equal to the
sum of the numbers of invariants of all the Sylow subgroups of
G. It is clear that the independent generators of G can be so
selected that their number has any arbitrary value from a to
m', but this number can have no other value. Moreover, G
cannot be generated by less than a cyclic subgroups even if
these subgroups are not independent.
Whenever the independent generators of G are so chosen
that the order of each of them is divisible by the orders of all
those which follow it, their number must be a, and when the
order of each is a power of a single prime their number must
be m', but it is not true that the independent generators can be
so arranged that the order of each is divisible by the order of all
those which follow it whenever the number of these generators
is a. The two numbers a and m' are only equal when the order
of G is a power of a single prime. Since the former method
leads to the smallest number of invariants it seems appro