Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Representations of the Gupta-Sidki group


Authors: D. S. Passman and W. V. Temple
Journal: Proc. Amer. Math. Soc. 124 (1996), 1403-1410
MSC (1991): Primary 20C07; Secondary 16S34, 20E08, 20F50
DOI: https://doi.org/10.1090/S0002-9939-96-03241-8
MathSciNet review: 1307556
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If $p$ is an odd prime, then the Gupta-Sidki group ${\mathcal G}_p $ is an infinite $2$-generated $p$-group. It is defined in a recursive manner as a particular subgroup of the automorphism group of a regular tree of degree $p$. In this note, we make two observations concerning the irreducible representations of the group algebra $K[{\mathcal G}_p ]$ with $K$ an algebraically closed field. First, when $\operatorname {char} K\neq p$, we obtain a lower bound for the number of irreducible representations of any finite degree $n$. Second, when $\operatorname {char} K=p$, we show that if $K[{\mathcal G}_p ]$ has one nonprincipal irreducible representation, then it has infinitely many. The proofs of these two results use similar techniques and eventually depend on the fact that the commutator subgroup ${\mathcal H}_p $ of ${\mathcal G}_p $ has a normal subgroup of finite index isomorphic to the direct product of $p$ copies of ${\mathcal H}_p $.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 20C07, 16S34, 20E08, 20F50

Retrieve articles in all journals with MSC (1991): 20C07, 16S34, 20E08, 20F50


Additional Information

D. S. Passman
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication: ExperTune Software, 4734 Sonseeahray Drive, Hubertus, Wisconsin 53033

W. V. Temple
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication: ExperTune Software, 4734 Sonseeahray Drive, Hubertus, Wisconsin 53033

DOI: https://doi.org/10.1090/S0002-9939-96-03241-8
Received by editor(s): November 8, 1994
Additional Notes: The first author’s research supported in part by NSF Grant DMS-9224662.
Communicated by: Lance W. Small
Article copyright: © Copyright 1996 American Mathematical Society