The Fixity of Permutation Groups

✍ Scribed by J. Saxl; A. Shalev

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 912 KB
• Volume: 174
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1995.1171

✦ Synopsis

The fixity of a finite permutation group (G) is the maximal number of fixed points of a non-trivial element of (G). We analyze the structure of non-regular permutation groups (G) with given fixity (f). We show that if (G) is transitive and nilpotent, then it has a subgroup whose index and nilpotency class are both (f)-bounded. We also show that if (G) is primitive, then either it has a soluble subgroup of (f)-bounded index and derived length at most 4 , or (F^{*}(G)) is (\operatorname{PSL}(2, q)) or (S z(q)) in the natural permutation representations of degree (q+1, q^{2}+1) respectively.

() 1945 Academic Press. Ins.