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

112

ABELIAN GROUPS

[CH. IV

group of order pm. If the order of an abelian group is not a
power of a single prime number its characteristic subgroups
are found by forming the direct product of the characteristic
subgroups of its Sylow subgroups, and all such direct products
are characteristic subgroups of G.
EXERCISES
1. If an abelian group G of order pm has only two distinct invariants
p. l p%2, and if al-a2-=n, then the number of the characteristic subgroups
which are generated by the operators of order pa is 2, when n= 1 and 6
has any one of the values from 1 to ai- 1. The number of these subgroups
cannot exceed the smaller of the two numbers n -1-1, a 2+1 for any value
of S.
2. Find all the characteristic subgroups of the abelian group of order
p6 and of type (1, 2, 3).
3. The abelian group of order 16 and of type (1, 3) has the property
that no single set of operators of order 4, which are conjugate under its
group of isomorphisms, generates all its operators of this order. Prove that
whenever p>2 all the operators of order p2 in the abelian group of order
p4 and of type (1, 3) are generated by a single set of operators of order p2
which are conjugate under its I.
45. Non-abelian Groups in which Every Subgroup is Abelian.
Let G represent any non-abelian group all of whose subgroups
are abelian. As instances of such groups we may cite the
octic and quaternion groups. We shall first prove that G
must contain an invariant subgroup of prime index p. Suppose
that G is represented as a transitive substitution group of the
smallest possible degree. If this group is imprimitive it must
transform a set of systems of imprimitivity according to a primitive group which has a (1, a) isomorphism with G.
This primitive group must be such that each of its subgroups
is abelian, and hence we have only to prove that every primitive group which contains only abelian subgroups has an
invariant subgroup of index p. If this primitive group were
regular it would be of order p. If it were non-regular and of
degree n, a maximal subgroup of degree n-1 would be abelian
and hence all of its substitutions besides the identity would be
of degree n-1; for, if the degree of such a substitution were
less than n-1, this substitution would occur in two maximal