# 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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## Subgroup growth in some pro-\$p\$ groupsHTML articles powered by AMS MathViewer

by Yiftach Barnea and Robert Guralnick
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
• J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-\$p\$ groups, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 61, Cambridge University Press, Cambridge, 1999. MR 1720368, DOI 10.1017/CBO9780511470882
• Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
• S. Minakshi Sundaram, On non-linear partial differential equations of the parabolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 479โ494. MR 0000088
• Alexander Lubotzky and Aner Shalev, On some \$\Lambda\$-analytic pro-\$p\$ groups, Israel J. Math. 85 (1994), no.ย 1-3, 307โ337. MR 1264349, DOI 10.1007/BF02758646
• A. Mann, Subgroup growth in pro-\$p\$ groups, in New Horizons in Pro-\$p\$ Groups, eds: M. du Sautoy et al., Progress in Mathematics 184, Birkhรคuser, Boston, 2000, pp. 233โ247.
• Aner Shalev, Growth functions, \$p\$-adic analytic groups, and groups of finite coclass, J. London Math. Soc. (2) 46 (1992), no.ย 1, 111โ122. MR 1180887, DOI 10.1112/jlms/s2-46.1.111
• I.O. York, The Group of Formal Power Series under Substitution, Ph.D. Thesis, Nottingham, 1990.
Similar Articles
• Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20E18, 17B70
• Retrieve articles in all journals with MSC (2000): 20E18, 17B70
• 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