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

104

ABELIAN GROUPS

[CH. IV

correspond to itself multiplied by the operator which corresponds
to it in this isomorphism.
The simplest case that can.present itself is the one in which
the subgroup of G which corresponds to the identity of T
includes T. The resulting simple isomorphism of G with itself
must correspond to an operator in the group of isomorphisms
of G, whose order is equal to the order of the operator of highest
order in T. When the order of T is an odd prime number p,
or double such a number, only one other case can present
itself; viz., the case in which T, or its subgroup of odd order,
corresponds to itself in the given isomorphism between G and T.
In this case the isomorphism corresponds to an operator whose
order divides p-1, in the group of isomorphisms of G. These
results give rise to the following theorem: If we make an abelian
group G simply isomorphic with itself by multiplying its operators by those of a subgroup whose order is p, or 2p (p being an odd
prime), the resulting automorphism of G corresponds to an operator of order p, 2p, or (p-1)/a (a being a divisor of p-I), in the
group of isomorphisms of G.
The determination of all the possible orders of the corresponding operators in the group of isomorphisms of any abelian
group, when T is a given subgroup, seems to be a problem of
considerable difficulty. When the order of T is small the number of cases that have to be considered is also small. In addition to the orders included in the given theorems, we have the
following, when the order of T does not exceed 8: If T is the
cyclic group of order four, the resulting isomorphism may also
correspond to an operator of order two in the group of isomorphisms, and when T is the non-cyclic group of this order, it may
also correspond to operators of orders 3 and 4. When T is the
cyclic group of order 8, the orders of these operators may be 2, 4,
and 8; when T is the direct product of the cyclic group of order
4 and an operator of order 2, the orders of the corresponding
operators in the group of isomorphisms may be 2 and 4; finally,
when T is the direct product of three operators of order 2, the
given operators may be of orders 2, 3, 4, 6, and 7. While all
of the possible cases for a given T may present themselves in