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

CHAPTER XII

THE LINEAR GROUPS IN THREE VARIABLES
111. Introduction. As in the case of the binary groups
we shall limit ourselves to the discussion of groups of transformations of determinant unity, and shall generally write
gc for the order of a linear group whose corresponding collineation group is of order g. Moreover, the order of a transformation S will often be written in the form go, when the order of
the group generated by S is gp. For instance, the order of the
transformation
S=(a, a, ac2) (a3 = co; 3 = )
may be written either 9 or 34.
Though the orders as written may thus virtually refer to
collineation groups, it must be kept in mind that all purely
descriptive terms refer to linear groups. For instance, the
group of order 90 generated by
0 1 0O
Sl=(l,(, W,2), S2= 0 0 1
1 0 0
would be described as a non-abelian group, though the corresponding collineation-group is abelian.
The determination of the linear groups in three variables
is based upon the following classification:
1. Intransitive and imprimitive groups.
2. Primitive groups having invariant imprimitive subgroups.
3. Primitive groups whose corresponding collineation groups
are simple.
4. Primitive groups having invariant primitive subgroups.
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