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

218 THE LINEAR GROUPS IN TWO VARIABLES [CH. X

Similarly we find
cos 2v sin 2v 0 1 0 0
S2S2= -sin 2v cos 2v 0, S3S3= 0 cos 2w -sin 2w
0 0 1 0 sin 2w cos 2w
If we interpret X, Y, Z as rectangular coordinates in ordinary space, we recognize here three real rotations around the
X-, Z-, X-axes respectively, the origin remaining fixed. The
rotations performed successively will, as is well known, be
equivalent to a single rotation. With the transformations
of the group G are therefore associated rotations which evidently form a group G' isomorphic with G. The isomorphism
is (1, 2) in the case where G contains E1=(-l, -1); otherwise it is (1, 1), since we may readily prove that to identity
of G' will correspond only E = (1, 1) or E1 of G. In other words,
G' is simply isomorphic with the collineation-group corresponding to G.
100. The Regular Polyhedron. Consider an axis of rotation (L) of G', and let the various angles of rotations around
L be the different multiples of (360/m)~; we shall say that
L is of index m. Let P1 be one of the points where L cuts the
sphere 2. This point will be transformed into (say) k distinct
points upon Z by G': Pi, P2,.., Pt, all of which will be
extremities of axes of rotation of index m. The distribution
of these points about any one of them is similar to the distribution about any other.
Now let arcs of great circles be drawn connecting Pi with
all the other points P2,..., Pk, and let the shortest arc be
of length A. The number of arcs of this length radiating
from P1 is m or a multiple of m, since always m of the arcs are
interchanged by rotations about L through the different multiples of (360/m)~. However, there cannot be more than 5 arcs
A; an exception occurring where we have just one or two
points P1, P2, one or both extremities of L, in which case A = 360~
or 180~. For, if there were 6 or more, a pair of them (say
C,, Ct) would make an angle 3 _~ 60~ with each other at P1;