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| Paper IPM / M / 17513 |
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| Abstract: | |
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Let G be a non-abelian group and
be the center of G. The non-commuting graph $\Gamma_G$ associated to G is the graph whose vertex set is $G\setminus Z(G)$ and two distinct elements are adjacent if and only if $xy\neq yx$ . We prove that if G and H are non-abelian groups with isomorphic non-commuting graphs, such that G is nilpotent, then H is nilpotent, provided $|Z(G)\geq |Z(H)||$ . Actually we prove conjecture 3 proposed in [V. Grazian and C. Monetta, A conjecture related to the nilpotency of groups with isomorphic non-commuting graphs, J. Algebra 633 (2023) 389-402]
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