Finitep′-Semiregular Groups

✍ Scribed by P. Fleischmann; W. Lempken; P.H. Tiep

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 349 KB
• Volume: 188
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.6858

✦ Synopsis

In doing so he also gave a full description of those finite groups which have a Ž . Ž . representation D over ‫ރ‬ such that D g has no eigenvalue 1 for any nontrivial element g g G. In our terminology such a group is called w x Ž . semiregular. One of the results in 17 characterizes SL 5 as the only 2 perfect semiregular group.

Clearly, if the underlying field ‫ކ‬ has characteristic p dividing the group order, then p-elements of G always act with eigenvalue 1 on any ‫ކ‬Gmodule. So in this situation the notion of ''semiregularity'' naturally modifies to a local version which in this paper is called '' pЈ-semiregularity'' Ž . cf. Definition 1.1 . Of course this localized semiregularity can also be studied in the complex case. In particular a finite group is semiregular if and only if it is pЈ-semiregular for all primes p.

So in this paper we generalize H. Zassenhaus' result and classify the finite pЈ-semiregular groups. Moreover we obtain group theoretical critew x ria for pЈ-semiregularity which generalize those given in 17 .

* Supported by the Alexander von Humboldt Foundation.