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

216 THE LINEAR GROUPS IN TWO VARIABLES [CH. X

98. Outline of the Process. 1~. Let G be a group in two
variables xi, x2. Then by the introduction of the conjugateimaginary group G (cf. ~ 92) and by the selection of new variables X, Y, Z which are bilinear in xl, X2 and their conjugateimaginary values xl, x2, we obtain a group G' of real rotations
in space, leaving the origin fixed (~ 99).
2~. Consider now a sphere Z of radius 1 whose center is the
origin. With each rotation of G' belongs an axis of rotation.
One of the points where such an axis pierces Z together with
all those points into which this point is moved by G' form the
vertices of a regular polyhedron, including the limiting cases
where there is a single axis of rotation or where the polyhedron
becomes a flat polygon (~ 100).
3~. The determination of G' is therefore made to depend
upon the construction of the analytical expressions representing the rotations of the regular solids. We find five different
types for G' and correspondingly five different types for the
linear groups G (~~ 101-103).
99. The Group of Rotations G'. Let
a b
S=l
Lc d
be any transformation of G, whose variables Xi, X2 are chosen
such that the Hermitian invariant is I =xlil- +X2X2 (Cor., ~ 92).
Then the following equations are true (~~ 93, 97):
ad-bc= 1 = ad-bc,
aa +bb=l, ac+ bd=O, cc+dd=l.
From these we obtain
c=-b, d=a.
Moreover, if we let p, q represent the positive square roots of
aa and bb respectively, and put a=pa, b=q3, we get, since
p=p, q=q:
c=-q3, d=pa;
aa=f3= 1=l, p2+q2 1.