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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 $p$ 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?)

  • [BG] A. D. Bell and K. R. Goodearl, Algebras of bounded finite dimensional representation type, Glasgow Math. J. (to appear).
  • [DZ] Z.Z. Dyment and A. E. Zalesskii, On the lower radical of a group ring, in Russian, Dokl. Akad. Nauk BSSR 19 (1975), 876--879. MR 52:5723
  • [F] D. R. Farkas, Semisimple representations and affine rings, Proc. Amer. Math. Soc. 101 (1987), 237--238. MR 88h:16027
  • [GS] N. Gupta and S. Sidki, On the Burnside problem for periodic groups, Math. Z. 182 (1983), 385--388. MR 85g:20075
  • [H] I. N. Herstein, Noncommutative Rings, Carus Math. Monograph #15, M.A.A., 1968. MR 37:2790
  • [KS] H. Kraft and L. W. Small, Invariant algebras and completely reducible representations, Math. Research Letters 1 (1994), 297--307. CMP 95:04
  • [P] D. S. Passman, The Algebraic Structure of Group Rings, Wiley-Interscience, New York, 1977. MR 81d:16001
  • [S1] S. Sidki, On a $2$-generated infinite $3$-group: subgroups and automorphisms, J. Algebra 110 (1987), 24--55. MR 89b:20081b
  • [S2] ------, A primitive ring associated to a Burnside $3$-group (to appear).

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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

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