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

~ 50]

NUMBER OF THEIR SUBGROUPS

127

to the various possible sets of independent generators of any
prime power group. As every maximal subgroup of a group
of order pm is of order pm-1, it results directly from the definition, that the 0-subgroup of any group of order pm is the crosscut of all its subgroups of index p. Hence the ~-subgroup of
a group of order pm can also be defined as its smallest invariant
subgroup which gives rise to an abelian quotient group of type
(1, 1, 1,... ). If the order of this quotient group is p", it
follows that a is the number of independent generators in
every possible set of such generators of this group of order pm.
In particular, every possible set of independent generators of any
prime power group involves the same number of operators. That is,
the number ot operators in each of the possible sets of independent generators of a Sylow group is an invariant of this group.
From the preceding paragraph it results directly that a
necessary and sufficient condition that the ~-subgroup of a
given group of order pm be the identity is that this group be the
abelian group of type (1, 1, 1,... ). Hence there is one
and only one group of order pm, p being any prime number
and m being any positive integer, which has the identity for
its ~-subgroup. The number of operators in every set of independent generators of this group is m. In every other group
of order pm the number of these independent generators is less
than m, and there is at least one group of order pm in which
this number is any arbitrary positive integer from 1 to m.
EXERCISES
1. The number of abelian subgroups of order p3 in any group is either
0 or of the form 1 +kp, even if the order of the group is not a power of p.
2. If an abelian group of order pm has p2 for one invariant while all
of its other invariants are equal to p, the number of its subgroups of order
p-1 i - 1
pm-i -s
p-1
3. A non-abelian group of order pS+l contains 0, 1, or p +1 abelian subgroups of order ps; in the last case it contains pS-l invariant operators.
4. If a group of order 2m contains only one subgroup of order 2 it is
either cyclic or dicyclic, and if a group of order p", p>2, contains only
one subgroup of order p it must be cyclic.*
* W. Burnside, Theory of Groups, 1911, p. 132.