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

220 THE LINEAR GROUPS IN TWO VARIABLES [CH. X

sequently, G' contains just one axis of index 2, or just three such
which are mutually perpendicular.
THE GROUPS OF THE REGULAR POLYHEDRA, ~~ 101-103
101. Limiting Cases. We notice first that the most general
linear homogeneous change of variables (xl, X2) in G is indicated
by a linear transformation T (~ 88) to which again corresponds
the most general rotation of Z about its center. It follows that
any given configuration arrived at in ~ 100 may at the outset
be placed in any required position relative to the axes of
coordinates X, Y, Z.
Beginning then with the simplest case where there.is a single
axis L of rotation, we let this be the X-axis. Then sin 2v =0
and cos2v=l (cf. ~ 99). Hence S has the form (~a, ~-a-l).
If S is of order g we have (=ia) =l.
(A) G': a single axis of index g;
G: an abelian group (intransitive) of order g:
SX=(EX, E-X); X=l, 2,..., g; e=l.
The next case to be considered is where there is an axis L of
index g, assumed to be the X-axis as above, in addition to g axes
of index 2 lying in a plane perpendicular to L. Let one of the
latter be the Z-axis; we then have cos 2v = -1, cos 2(u-w)= 1,
and the corresponding transformation of G is found to be
T =(B) Dihedral Group.
G': one axis L of index g and g axes of index 2;
G: an imprimitive group of order 2g~ consisting of the transformations
0 dSx=(=t=x, eX-X), Tx =; X=l, 2,..., g; e~=1..=Fe-X 0