Note on structural properties of graphs

Abstract

Abstract In this paper, we establish sufficient and necessary conditions for the existence of abelian subgroups of maximal order of a finite group G G , by means of its commuting graph. The order of these subgroups attains the bound c = ∣ [ x 1 ] ∣ + ⋯ + ∣ [ x m ] ∣ c=| \left[{x}_{1}]| \hspace{-0.25em}+\cdots +\hspace{-0.25em}\hspace{0.33em}| \left[{x}_{m}]| , where [ x i ] \left[{x}_{i}] denotes the conjugacy class of x i {x}_{i} in G G and m m is the smallest integer j j such that ∣ [ x 1 ] ∣ + ⋯ + ∣ [ x j ] ∣ ≥ ∣ C G ( x j ) ∣ | \left[{x}_{1}]| \hspace{-0.25em}+\cdots +| \left[{x}_{j}]| \ge | {C}_{G}\left({x}_{j})| , where C G ( x j ) {C}_{G}\left({x}_{j}) is the centralizer of x j {x}_{j} in G G .