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

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NUMBER OF NON-CYCLIC SUBGROUPS

133

every group which contains only one cyclic subgroup of order
2" contains an odd number of cyclic subgroups of order 4. For
each value of a and m there are three such groups; hence there
is an infinite system of groups of order 2m which contain an odd
number of cyclic subgroups of composite order. When m=3
there are only two groups having this property, viz., the
quaternion group and the group of movements of the square.
If all the non-cyclic groups of order pm(m>3, p an arbitrary
prime) were determined, there would be just three among them
in which the number of cyclic subgroups of composite order
would not always be a multiple of p. In.these three special
cases the number of cyclic subgroups of every composite order
is not divisible by p.
In the exceptional groups noted above the number of the
subgroups of order 2 is =1 (mod 4). That this number is
3 (mod 4) in every other non-cyclic group of order 2m is a
direct consequence of the fact that the number of cyclic subgroups of order 4 in all these groups is even. From this fact
it results that the number of operators whose order exceeds
2 is divisible by 4, since every cyclic subgroup of order 2' contains 2~-1 operators which are not found in any other subgroup
whose order is - 2'. Hence the given system of groups is
composed of all the groups of order pm in which the number
of subgroups of order p is not l+p (mod p2). That is, the
groups of order pm in which the number of cyclic subgroups of
composite order is not divisible by p coincide with those in which
the number of subgroups of order p is not of the form 1 +p +kp2.
EXERCISES
1. The number of subgroups of order p in any non-cyclic group of order
pm, p>2, is of the form 1+p+kp2.
Suggestion: Consider the forms of the number of the operators of orders
p2, p3, etc.
2. The number of cyclic groups of order pa, p>2 and 3> 1, is of the form
kp whenever the Sylow subgroups of order pm are non-cyclic.
3. If a group G of order pm contains exactly p cyclic subgroups of order
p", these subgroups generate a characteristic subgroup of order pa+l
under G, and this subgroup is either abelian and of type (a, 1) or it is the
non-abelian group which is conformal with this abelian group.