Representations of the Gupta-Sidki group
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- by D. S. Passman and W. V. Temple PDF
- Proc. Amer. Math. Soc. 124 (1996), 1403-1410 Request permission
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
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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
- MR Author ID: 136635
- 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
- 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
- © Copyright 1996 American Mathematical Society
- 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