An Introduction to the Theory of Groups by Paul Alexandroff

Author:Paul Alexandroff
Language: eng
Format: epub, mobi
Publisher: Dover Publications, Inc
Published: 2013-06-26T16:00:00+00:00

Fig. 9

(b) Three subgroups of order eight which are isomorphic to the group of the square double pyramid. Each of these subgroups consists of those rotations of the cube which carry over into itself one of the lines joining the centroids of two opposite faces, for example the points S and S′. (The octahedron inscribed in the cube is a special case of the square double pyramid. The group of those of its rotations leaving fixed two of its vertices S and S′ or interchanging them is evidently identical with the group of the square double pyramid.)

Such a subgroup of order eight consists of the following eight rotations: Four rotations about the axis SS′ (including the identity); two rotations through the angle about the axes joining respectively the mid-points of the edges AA′ and CC′, and BB′ and DD′; and two rotations through the angle about the axes joining respectively the centroids of the faces ABB′A′ and CDD′C′, and ADD′A′ and BCC′B′.

(c) A subgroup of order four which consists of the identical rotation and of three rotations through the angle about the axes joining the centroids of two opposite faces. This group consists of those rotations which occur in each of the three subgroups of order eight mentioned above. It is commutative and isomorphic to the rotation group of the rhombus, and therefore also to Klein’s four-group.

The group of congruences or rotations of a regular octahedron is isomorphic to the rotation group of a cube.

In order to convince ourselves of this it is sufficient to describe round the regular octahedron a cube (fig. 10) or equally well to inscribe in the regular octahedron a cube (fig. 11). To each congruence of the octahedron there corresponds a certain congruence of the cube, and conversely.

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