To discuss the properties of a group of finite order some concrete form of representation of the group is necessary, except perhaps in the simplest cases. What are called the abstract defining relations (viz., a system of relations of the form A
i
= 1, B
j
= 1, . . . . . . A
p
B
q
. . . = 1, A
p'
B
q'
. . . = 1, . . . . . between a system of non-commutative symbols A, B, . . . , which are necessary and sufficient to ensure that only a finite number of distinct products can be formed from them) no doubt contain implicitly in the most concise form all the properties of the group. To establish the properties, however, on this basis is not in general practicable. For every group there are an infinite variety of possible concrete representations; and in general for an adequate discussion of the properties of the group several of them have to be made use of. In a limited class of cases, including, however, several groups of great importance in analysis, a representation as a group of space-collineations is available. In all such cases it would be expected that this form of representation, as affording scope for space-intuition, would certainly be one of those chosen for discussion. Except, however, as regards the so-called groups of the regular solids,
i. e.
, groups of rotations round a point, this has not been done. It is proposed in this memoir to discuss the simple group of order 25920 entirely from the point of view of projective geometry. The existence of a group of collineations of this order is not assumed, but is shown to follow from the existence of a remarkable configuration of points, lines and planes in space. This configuration itself arises naturally in connection with a much less complex group of space-collineations. The method followed throughout is synthetical and constructive. To avoid unduly burdening the earlier part of the memoir, it is assumed that the projective groups of finite order on the straight line have been established (as they can be) without appeal to analysis. Further, the simpler properties of the permutation-groups of 4, 5 and 6 symbols are taken as known.
Note on the defining relations for the simple group of order 660