Volume 32 • Number 1 • 2009   • On Finite Groups with Some Conditions on Subsets Bijan Taeri Abstract. Let be a positive integer. We denote by the class of groups such that, for every subset of of cardinality , there exist a positive integer , and a subset , with and a function , with and non-zero integers such that , where , , and whenever , for some subgroup of . If the integer is fixed for every subset we obtain the class . If one always has , , and , , one obtains the class . In this paper, we prove that   (1) A finite semi-simple group has the property , for some , if and only if or ,   (2) A finite non-nilpotent group has the property , for some , if and only if ,        where is the hypercenter of ,   (3) A finite semi-simple group has the property , for some , if and only if , where and denote the alternating and symmetric groups of degree n respectively. 2000 Mathematics Subject Classification: 20F99, 20F45. Full text: PDF