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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Subgroup growth in some pro-$p$ groups


Authors: Yiftach Barnea and Robert Guralnick
Journal: Proc. Amer. Math. Soc. 130 (2002), 653-659
MSC (2000): Primary 20E18; Secondary 17B70
Published electronically: August 29, 2001
MathSciNet review: 1866015
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Abstract:

For a group $G$ let $a_{n}(G)$ be the number of subgroups of index $n$and let $b_{n}(G)$ be the number of normal subgroups of index $n$. We show that $a_{p^{k}}(SL_{2}^{1}(\mathbb{F}_{p}[[t]])) \le p^{k(k+5)/2}$ for $p>2$. If $\Lambda=\mathbb{F}_{p}[[t]]$ and $p$ does not divide $d$or if $\Lambda=\mathbb{Z}_{p}$ and $p \ne 2$ or $d \ne 2$, we show that for all $k$ sufficiently large $b_{p^{k}}(SL_{d}^{1}(\Lambda))=b_{p^{k+d^{2}-1}}(SL_{d}^{1}(\Lambda))$. On the other hand if $\Lambda=\mathbb{F}_{p}[[t]]$ and $p$ divides $d$, then $b_{n}(SL_{d}^{1}(\Lambda))$ is not even bounded as a function of $n$.


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Additional Information

Yiftach Barnea
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706
Email: barnea@math.wisc.edu

Robert Guralnick
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-1113
Email: guralnic@math.usc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06099-3
PII: S 0002-9939(01)06099-3
Received by editor(s): March 1, 2000
Received by editor(s) in revised form: September 18, 2000
Published electronically: August 29, 2001
Additional Notes: Both authors wish to thank MSRI for its hospitality. The second author was also partially supported by an NSF grant.
Communicated by: Lance W. Small
Article copyright: © Copyright 2001 American Mathematical Society