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

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Abstract: If is an odd prime, then the Gupta-Sidki group is an infinite -generated -group. It is defined in a recursive manner as a particular subgroup of the automorphism group of a regular tree of degree . In this note, we make two observations concerning the irreducible representations of the group algebra with an algebraically closed field. First, when , we obtain a lower bound for the number of irreducible representations of any finite degree . Second, when , we show that if 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 of has a normal subgroup of finite index isomorphic to the direct product of copies of .

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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