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

14

EXAMPLES OF GROUPS; DEFINITIONS

[CH. i

It is not difficult to verify that the following six matrices
also constitute a group of order six -as regards multiplication:
1 0 -1. 1 0 - 1 0 1 00 - 0 1
01 -1 0 I' 1-1' 0 -1 1 0
This group of order six can, however, not be put in a (1, 1)
correspondence with the one considered in the preceding paragraph, although the first three matrices are the same in the
two groups. The present group of order six is generated by a
single matrix and hence it is cyclic. As a generating matrix
we may use either one of the last two matrices. It may also
be observed that multiplication is commutative in the present
group, while this is not the case in the group of the preceding
paragraph. The matrices of the present group can be placed
in a (1, 1) correspondence with the six sixth roots of unity as
regards the operation multiplication.
The four matrices
110 -10 1 0 -1 0
01' 0 -1 0 -1 01
evidently serve as another illustration of the non-cyclic group
of order 4 when they are combined by multiplication, while
the four matrices
10 i-10 01 0-1
0 1' 0 —1' -1 ' 1 0
combined according to the same operation, constitute the cyclic
group of order 4. As an instance of matrices which constitute
a cyclic group of an arbitrary order we may observe that the
matrices
10 11 12 13 1 m-1
01'0 i 0?0 01'' 0 1
constitute the cyclic group of order m when they are combined
as to multiplication, and the elements are reduced modulo m.