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

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SEVERAL USEFUL THEOREMS

171

of H while the operators of H are left unchanged, then the group
of isomorphisms of G involves an invariant subgroup which is
simply isomorphic with H, whenever this group of isomorphisms
can be represented as a transitive group of degree h, corresponding
to the conjugates of s.
Suppose that G is a complete group which involves only
one subgroup of index 2, and consider the direct product of G
and the group of order 2. If a group contains only one subgroup of index 2, this subgroup is generated by the square of
the operators of the group, and, conversely, if a subgroup of
index 2 is generated by the squares of the operators of a
group, it is the only subgroup of index 2 in the group. That
is, a necessary and sufficient condition that a group contain
one and only one subgroup of index 2 is that the squares of its
operators generate such a subgroup. Hence the squares of the
operators in the direct product of G and the group of order
2 generate a characteristic subgroup of index 4 under this
direct product. The I of this product involves an invariant
operator of order 2 corresponding to the automorphisms in
which two of the three co-sets as to the given characteristic
subgroup are multiplied by the invariant operator of order 2.
As the order of this I is the double of the order of G and as I
involves an invariant operator of order 2 which is not in the I
of G there results the theorem:
If G is a complete group and contains only one subgroup of
index 2, then the group of isomorphisms of the direct product of G
and the group of order 2 is simply isomorphic with this direct
product.
As special cases of this theorem we may observe that the I
of the direct product of the symmetric group of degree n, n> 2
and n#6, and the group of order 2, is simply isomorphic with
this direct product; and that the I of the direct product of the
metacyclic group of order p(p-1), p being an odd prime,
and the group of order 2, is simply isomorphic with this direct
product. If a group G is the direct product of characteristic
subgroups, the I of G is evidently the direct product of the
I's of these subgroups. As an abelian group is the direct prod