Matrix Representations of Groups by Morris Newman

Author:Morris Newman
Language: eng
Format: epub
Publisher: Courier Publishing
Published: 2019-09-02T16:00:00+00:00

where ξ, η, are any pth roots of unity; and the p −1 irreducible representations of degree p may be taken as

where r may have any of the values 1, 2, . . . , p −1.

When p is even there is also just one other nonabelian group of order p3 = 8, namely the quaternion group. This has previously been discussed.

(d) Let G be a finite group of odd order h = 2t + 1. We shall show that the only irreducible representation of G which is real is the principal representation, and that conjugate irreducible representations not equivalent to the principal representation are themselves inequivalent. (The principal representation is the representation of degree 1 such that every group element is assigned the value 1.)

Let α be any irreducible representation of G not equivalent to the principal representation α1. Let m be the degree of α. Then m is odd, since m|h. Since h is odd, the group elements may be written as

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