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

CHAPTER X

THE LINEAR GROUPS IN TWO VARIABLES
97. Introduction. We shall limit ourselves to the determination of the groups whose transformations have unity for
the value of their determinants. From these all other forms
of groups may readily be constructed (cf. ~~ 83-87). We
shall say that a given group is a type of all groups which may be
obtained from it by a mere change of variables.
All the types we encounter (with one exception) contain
a group of similarity-transformations of order 2, E=(1, 1),
El=(-1, -1), due to the fact that a linear transformation
of determinant unity and of order 2 does not exist unless it
be the similarity-transformation E1. A transformation whose
corresponding collineation is of order 2, if written in canonical
form, must necessarily be (i, -i), where i2= -1. The exception mentioned is the type of an abelian group of odd order.
Following Jordan, it shall be our practice to call the order of
a linear group go, if g is the order of the corresponding collineation-group, and 0 the order of the subgroup of similarity-transformations contained in the given group.
There are several processes available for the determination
of the types of groups sought.* We shall here employ a modified form of Klein's original process, which depends largely on
geometrical intuition.
* Klein, Mathematische Annalen, Bd. 9 (1876), p. 183 ff.; Vorlesungen iiber
das Ikosaeder, Leipzig, 1884, pp. 116-120. Gordan, Mathematische Annalen, Bd.
12 (1877), p. 23 ff. Jordan, Journal fur die reine und angewandte Mathematik,
Bd. 84 (1878), pp. 93-112; Atti della Reale Academia di Napoli, t. 8 (1879).
Fuchs, Journal fiir die reine und angewandte Mathematik, Bd. 81, 85 (1876, 1878),
pp. 97, 1 ff. Valentiner, De endelige fransformations-gruppers Theori, Copenhagen, 1889, p. 100 ff.
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