A Study of Certain Substitution Groups

This thesis purposes to study a certain group of movements which can be expressed as substitutions. The groups of movements which send a square into itself is to be studied as a group of eight substitutions on the vertices for the purpose of leading up to the real problem of this paper. From the octic group, it is natural to proceed to a study of the movements which send a cube into itself. In particular, it is the aim of this thesis to discover the group of the cube and to analyze some of its properties. There are twenty-eight rotations and reflections with respect to diagonals and central axes of the cube which possess special geometrical properties. One of the problems of this thesis is to determine whether or not these twenty-eight elements constitute a group. Once the group of the cube has been determined, other problems are those of finding subgroups within the original group and of enumerating their properties. This paper is to be concerned chiefly with subgroups composed entirely of elements from twenty-eight rotations and reflections with the special geometrical properties. Also a few theorems relative to groups in general will be demonstrated and application will be made to the group of the cube