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

70

ABSTRACT GROUPS

[CH. III

verse is not necessarily true. That is, there are perfect groups
which are not also simple, as we shall see later.
If the elements of a commutator belong to two invariant
subgroups of a group G, this commutator is contained in the
cross-cut of these invariant subgroups. Hence it results
that if two invariant subgroups of G have only the identity in common, every element of each one of these subgroups is commutative
with every element of the other.
If the elements of the commutator s-1t-lst are permuted
in every possible manner, there result eight operators which may
be distinct and may all differ from the identity. These eight
operators are: s-~t-~st, t-1sts-1, sts-'t-1, ts-1t-'s, t-'s-~ts,
st-ls-1t, tst-ls-1, s-~tst-1. All of them can be obtained from
any one of them by means of the substitution group of order 8
on the four factors. It is evident that each of these 8 commutators has the same order.
To prove this it may first be observed that the order of any
product of n operators is invariant as regards the cyclic group oj
permutations of these factors, since such permutations are equivalent to transforming by elements. If reversing the order of
these n factors does not affect the order of the product, this
order is invariant as regards the dihedral group of permutation
of its n factors. In particular, the order of the product of n elements of order 2 is always invariant as regards the dihedral group
of permutations of the n elements. Reversing the order of the
factors of a commutator cannot affect the order of this commutator, since it is equivalent to a cyclic permutation of the factors of its inverse.
The given eight commutators, involvings, t and their inverses, are contained in the commutator subgroup of the group
generated by s and t, but they do not necessarily generate this
subgroup. Since four of them are the inverses of the other
four, it is clear that no more than four of them are distinct
when their common order is 2. The most general definition
of a commutator is, "the product of the transform of an element
and its inverse." Whenever an element can be written as such
a product, it may be called a commutator. When we speak