## 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

## 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 $n$.## References

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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
- MR Author ID: 78455
- Email: guralnic@math.usc.edu
- 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: https://doi.org/10.1090/S0002-9939-01-06099-3
- MathSciNet review: 1866015