# The size of the largest conjugacy classes and the Sylow p -subgroups of finite groups

Research paper by Yong Yang

Indexed on: 03 Dec '16Published on: 10 Nov '16Published in: Archiv der Mathematik

#### Abstract

Abstract Let p be a prime and let P be a Sylow p-subgroup of a finite nonabelian group G. Let bcl(G) be the size of the largest conjugacy classes of the group G. We show that if p is an odd prime but not a Mersenne prime or if P does not involve a section isomorphic to the wreath product $${Z_p \wr Z_p}$$ , then $${ P/O_p(G) \leq bcl(G)}$$ .AbstractLet p be a prime and let P be a Sylow p-subgroup of a finite nonabelian group G. Let bcl(G) be the size of the largest conjugacy classes of the group G. We show that if p is an odd prime but not a Mersenne prime or if P does not involve a section isomorphic to the wreath product $${Z_p \wr Z_p}$$ , then $${ P/O_p(G) \leq bcl(G)}$$ .pPpGGGpP $${Z_p \wr Z_p}$$ $${Z_p \wr Z_p}$$ $${ P/O_p(G) \leq bcl(G)}$$ $${ P/O_p(G) \leq bcl(G)}$$