Skip to main content Accessibility help
×
Home

THE NUMBER OF PROFINITE GROUPS WITH A SPECIFIED SYLOW SUBGROUP

  • COLIN D. REID (a1)

Abstract

Let $S$ be a finitely generated pro- $p$ group. Let ${\mathcal{E}}_{p^{\prime }}(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects nontrivially with every nontrivial normal subgroup of $G$ . In this paper, we investigate whether or not there is a bound on $|G:S|$ for $G\in {\mathcal{E}}_{p^{\prime }}(S)$ . For instance, we give an example where ${\mathcal{E}}_{p^{\prime }}(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $|G:S|$ is bounded in the case where $S$ is just infinite.

Copyright

References

Hide All
[1]Aschbacher, M., Finite Group Theory (Cambridge University Press, New York, 2000).
[2]Craven, D. A., Fusion Systems: An Algebraic Approach (Cambridge University Press, Cambridge, 2011).
[3]Feit, W. and Thompson, J. G., ‘Solvability of groups of odd order’, Pacific J. Math. 13 (1963), 7751029.
[4]Gagola, S. M. Jr. and Isaacs, I. M., ‘Transfer and Tate’s theorem’, Arch. Math. (Basel) 91(4) (2008), 300306.
[5]Gilotti, A. L., Ribes, L. and Serena, L., ‘Fusion in profinite groups’, Ann. Mat. Pura Appl. (4) 177 (1999), 349362.
[6]Leedham-Green, C. R. and McKay, S., The Structure of Groups of Prime Power Order (Oxford University Press, New York, 2002).
[7]Reid, C. D., ‘Finiteness properties of profinite groups’, PhD Thesis, University of London, 2010.
[8]Reid, C. D., ‘The generalised pro-fitting subgroup of a profinite group’, Comm. Algebra 41(1) (2013), 294308.
[9]Rose, J. S., A Course on Group Theory (Cambridge University Press, Cambridge, 1978).
[10]Stancu, R. and Symonds, P., ‘Fusion systems for profinite groups’, 2012, arXiv:1204.2582.
[11]Stather, M., ‘Constructive Sylow theorems for the classical groups’, J. Algebra 316(2) (2007), 536559.
[12]Symonds, P., ‘On cohomology isomorphisms of groups’, J. Algebra 313(2) (2007), 802810.
[13]Tate, J., ‘Nilpotent quotient groups’, Topology 3(Suppl. 1) (1964), 109111.
[14]Yoshida, T., ‘Character-theoretic Transfer’, J. Algebra 52 (1978), 138.
[15]Zassenhaus, H., ‘Beweis eines Satzes über diskrete Gruppen’, Abh. Math. Semin. Univ. Hambg. 12 (1938), 289312.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

THE NUMBER OF PROFINITE GROUPS WITH A SPECIFIED SYLOW SUBGROUP

  • COLIN D. REID (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed