ω-Groups and upperbound to the number of centralizers of non-central elements of finite groups

ω-Groups and upperbound to the number of centralizers of non-central elements of finite groups

Let G be a non-abelian group. The non-commuting graph TG of G is dened as the graph whose vertex set is the non-central elements of G and two vertices are joint if and only if they do not commute. In a finite simple graph TG the maximum size of a complete subgraph of 􀀀G is called the clique number of TG and it is denoted by ω(G), here it is the maximum cardinality of a pairwise non-commuting subset of G, and Cent(G) is used to denoted the set of centralizers of single elements of G. In this paper we determine the clique number of 􀀀G in new way. Also we determine that if S3 be dropped out of finite groups, then Prcent(G) ∈ (0,3/4) where Prcent(G) = Cent(G)/G and the Cent(G) is the number ofdistinct centralizers of non-central elements of G. Also we introduce groups that their clique numbers are the same as those of their central quotients.