| Summary: | In this paper, we establish sufficient and necessary conditions for the existence of abelian subgroups of maximal order of a finite group GG, by means of its commuting graph. The order of these subgroups attains the bound c=∣[x1]∣+⋯+∣[xm]∣c=| \left[{x}_{1}]| \hspace{-0.25em}+\cdots +\hspace{-0.25em}\hspace{0.33em}| \left[{x}_{m}]| , where [xi]\left[{x}_{i}] denotes the conjugacy class of xi{x}_{i} in GG and mm is the smallest integer jj such that ∣[x1]∣+⋯+∣[xj]∣≥∣CG(xj)∣| \left[{x}_{1}]| \hspace{-0.25em}+\cdots +| \left[{x}_{j}]| \ge | {C}_{G}\left({x}_{j})| , where CG(xj){C}_{G}\left({x}_{j}) is the centralizer of xj{x}_{j} in GG.
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