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

54

ABSTRACT GROUPS

[CH. III

equivalence remains true when m is any negative integer. In
particular, two elements of G which satisfy the equation
SaS3= 1
are said to be the inverses of each other, and s = sa- is denoted
by s-~l. Elements of order 2 are their own inverses; but all
other elements, besides the identity, go in pairs, composed
of an element and its inverse. In particular, every possible
group contains an even number of elements, which may be zero,
of every order which exceeds 2.
Since
sas s... s* s-1... - lS, =
it results that the inverse of ss#... sx is sx-l... s-ilsal,
and that
(SaSs... S^x)-'1 (Sx.., ~S -1 1)1
If all of the elements sa, so,..., sx are of order 2, then
(SaS.... sx)- x.. S. ss.
In particular, if the product of two elements, s,, so of order 2 is
also of order 2 the elements are commutative, that is sst =s0s,.
If the elements Sa, s,2,..., s,= 1 do not include all the
elements of G, they represent a subgroup of G. By exactly the
same arguments as were used to prove that the order of a substitution group is divisible by the order of each of its subgroups
(~ 9), it can be proved that the order of an abstract group is
divisible by the order of each one of its subgroups. Hence it
results that k is a divisor of g.
23. The Cyclic Group.* By definition the cyclic group
is generated by a single element, and every group which can be
generated by a single element is cyclic. If the order of such
a group is g, it contains at least one element s of the order g.
The order of sm, m being an arbitrary positive integer, is g/d,
d being the greatest common divisor of m and g. If q(g) repre* Many of the results of this section can be deduced from properties of the
n roots of unity. In fact, these n roots form a cyclic group with respect to multiplication.