Vol. 28, No. 2, 1969

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Finite groups in which every element is conjugate to its inverse

J. Lennart (John) Berggren

Vol. 28 (1969), No. 2, 289–293

Let S denote the class of all finite groups all of whose irreducible characters over C (the complex numbers) are real. It is easy to verify, but important to observe, that this condition is equivalent to the condition that every element of the group is conjugate to its inverse (under an inner automorphism). Since Sn S, where Sn denotes the symmetric group on n letters, any finite group may be embedded in a group in S. The goal of §1 will be to show that the two-Sylow subgroups of Sn also are in S, and, if An denotes the alternating group on n letters, that An S if and only if n ∈{1,2,5,6,10,14}. The result on the two-Sylow subgroups of Sn will be used to show that any finite two-group is embeddable in a two-group in S. G. A. Miller has studied a class of groups related to those in S. The main theorem of §2 gives a more intuitive characterization of the class of groups investigated by Miller, a consequence of which is a necessary and sufficient condition for a group in this class to be a member of S.

Mathematical Subject Classification
Primary: 20.25
Received: 25 October 1967
Published: 1 February 1969
J. Lennart (John) Berggren