Session 14. Group Rings and Related Topics

Sylow numbers of finite groups

Iris Köster, Universität Stuttgart, Germany
Let \(G\) be a finite group and let \(n_p(G)\) denote the number of Sylow \(p\)-subgroups of \(G\). The set \(sn(G)\) of all Sylow numbers \(n_p(G)\) is called Sylow numbers of \(G\). There are many results concerning the structure of groups with given Sylow numbers, especially about the solvability of groups with given Sylow numbers.

F. Luca showed in [1] based on a result of J.Zhang [2] that \(G\) is solvable provided \(|sn(G)|=2\), \(sn(G)=\lbrace 1,a,b\rbrace\) or \(sn(G)=\lbrace q^x,a,b\rbrace\), where \(q\) is a prime number and either \(\gcd (a,b)=1\) or \(q\not| \, ab\).

N. Chigira however showed that Zhangs's result is not valid for all groups [3]. 2012 A. Moreto gave a complete but different proof of the first part of Luca's claim [4].

In the first part of the talk the remaining parts of Luca's result are proved [5].

In the second part the question posed by A. Moreto in [4] whether Sylow numbers are determined by the character table of a group is considered. It is shown that this is the case when \(G\) is supersolvable (this reports on joint work with

References
  1. F. Luca, Groups with two Sylow numbers are solvable , Arch. Math. 71, 1998, 95-96.
  2. J. Zhang, Sylow Numbers of Finite Groups , Journal of Algebra 176, 1995, 111-123.
  3. N. Chigira, Number of Sylow subgroups and \(p\)-Nilpotence of finite groups , Journal of Algebra 201, 1998, 71-85.
  4. A. Moreto, Groups with two Sylow numbers are the product of two nilpotent Hall subgroups , Arch. Math 99, 2012, 301-304.
  5. I. Köster, Finite groups with Sylow numbers \(\{q^x,a,b\} \), Stuttgarter Mathematische Berichte 2014/10. (http://www.mathematik.uni-stuttgart.de/preprints/downloads/2014/2014-010.pdf)
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