Vol. 57, No. 2, 1975

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On groups with a single involution

Jerry Malzan

Vol. 57 (1975), No. 2, 481–489

This paper is concerned with the “ordinary” (over the complex numbers) representation theory of finite groups and in particular with matrix groups of the first and second kinds (that is, matrix groups which are similar to real groups or, alternatively, have real character but are not similar to real groups. In the event that the character is non-real, we speak of the third kind.)

The purpose of this paper is associate groups with exactly one involution with representations of the second kind, and this we do in two ways: First, by showing that any group possessing an irreducible representation of the second kind involves a non-trivial group with only one involution. Second, by showing that a group with only one involution cannot have a faithful irreducible representation of the first kind. It is well and long known that groups of odd order possess nontrivial irreducible representations of the third kind only, so that evenness of order is a necessity if matrix groups of the first or second kind are to be dealt with.

Mathematical Subject Classification 2000
Primary: 20C15
Received: 10 July 1974
Published: 1 April 1975
Jerry Malzan