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

8

EXAMPLES OF GROUPS; DEFINITIONS

[CH. I

A substitution whose powers give all the substitutions of a
group is said to generate this cyclic group, and the order of
this cyclic group is equal to the order of the substitution. It is
always possible to select such a generating substitution in
more than one way except when the order of the cyclic group
is 1 or 2. For instance, the cyclic group of order 4
1, ac bd, abcd, adcb
is generated by abcd as well as by adcb, and the cyclic group
of order 3
1, abc, acb
has also two generating substitutions.
In general, a substitution group is said to be generated by
a set of substitutions provided all of the substitutions of the
group can be obtained by combining those of the set. The least
number of substitutions that can generate a non-cyclic group
is two. Each of the non-cyclic groups which have been considered thus far can be generated by two of its substitutions.
For instance, the symmetric group of order 6 can be generated
by any one of its three possible pairs of two distinct substitutions of order 2. It can also be generated by any one of the
six distinct pairs composed of one substitution of order 2 and
one of order 3. Hence the symmetric group of order 6 has
nine distinct pairs of generating substitutions.
The octic group cannot be generated by every possible pair
of distinct substitutions of order 2, since some such pairs generate only four substitutions. In fact, it is easy to verify that
only four out of these ten possible pairs generate this group,
while each of the remaining six generate a group of order 4.
The square of the substitutions of order 4 cannot be used as
one of a pair of generating substitutions of the octic group,
but every other substitution besides the identity of this group
occurs in such a pair. Hence it is not difficult to verify that
there are exactly 12 possible pairs of generating substitutions of the octic group.
Any set of substitutions on n letters generates some substitution group on these letters, which is contained in the sym