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

68

ABSTRACT GROUPS

[CH. III

jugates of G1 under G are also conjugates under G2. This is,
however, impossible, when I1 and H-2 are conjugate ince the
elements of G2 cannot transform Gi into G2. As the assumption that p'=p has led to an absurdity, it has been proved
that the index of the cross-cut of any two distinct conjugate
subgroups under one of these subgroups is ahclays less than the
index of these subgroups under the entire group.
To find a simple illustration of this fundamental theorem,
suppose that the index of G1 under G is 2. It follows then
directly from this theorem that if G1 and G2 are two conjugate subgroups, the cross-cut of G1 and G2 must have an index
which is less than 2 under each of these subgroups. Hence
this index is 1, and GI, G2 are identical. Hence the given
theorem includes as a special case the theorem that a subgroup of index 2 under any group is invariant. This fact can
also be readily proved in other ways. Cf. preceding Exercises.
If the invariant subgroup H is composed of all the invariant
elements of G it is called the central of G and its complementary
group is known as the central quotient group of G. When this
central quotient group is abelian G is said to be metabelian.*
The central quotient group is also called the group of congredient isomorphisms of G. It is clear that the central of G
is always abelian. For instance, 1, -1 constitute the central
of the quaternion group, and the central of an abelian group
coincides with the group. In a non-abelian group the order of
the central cannot exceed the order of the group divided by 4,
and if this is the order of the central of G, the central quotient
group is the axial group. It is easy to prove that the central
quotient group is always non-cyclic.
29. Commutators, Commutator Subgroup, and the q4-subgroup. The element or operator t s-1t-~st is called the commutator of s and t, while its inverse is the commutator of t
and s. When s and t are commutative their commutator is
* W. B. Fite, Proceedings of the American Association for the Advancement
of Science, vol. 49 (1901), p. 41.
t The elements of a group are also called operators or operations. We shall
hereafter use the terms element, operator, and operation interchangeably, since
all of these terms are commonly found in the modern literature of group theory.