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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Subgroup growth in some pro-$p$ groups
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by Yiftach Barnea and Robert Guralnick PDF
Proc. Amer. Math. Soc. 130 (2002), 653-659 Request permission


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:
  • Robert Guralnick
  • Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-1113
  • MR Author ID: 78455
  • Email:
  • 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
  • © Copyright 2001 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 130 (2002), 653-659
  • MSC (2000): Primary 20E18; Secondary 17B70
  • DOI:
  • MathSciNet review: 1866015