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On the relation between the non-commuting graph and the prime graph | ||
| International Journal of Group Theory | ||
| مقاله 4، دوره 1، شماره 1، خرداد 2012، صفحه 25-28 اصل مقاله (256 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2012.469 | ||
| نویسندگان | ||
| Neda Ahanjideh* ؛ A. Iranmanesh | ||
| چکیده | ||
| $\pi(G)$ denote the set of prime divisors of the order of $G$ and denote by $Z(G)$ the center of $G$. The\textit{ prime graph} of $G$ is the graph with vertex set $\pi(G)$ where two distinct primes $p$ and $q$ are joined by an edge if and only if $G$ contains an element of order $pq$ and the \textit{non-commuting graph} of $G$ is the graph with the vertex set $G-Z(G)$ where two non-central elements $x$ and $y$ are joined by an edge if and only if $xy \neq yx$. Let $ G $ and $ H $ be non-abelian finite groups with isomorphic non-commuting graphs. In this article, we show that if $ | Z ( G ) | = | Z ( H ) | $, then $ G $ and $ H $ have the same prime graphs and also, the set of orders of the maximal abelian subgroups of $ G $ and $ H $ are the same. | ||
| کلیدواژهها | ||
| Non-commuting graph؛ Prime graph, Maximal abelian subgroups, Maximal independent set of the graph | ||
| مراجع | ||
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J.S. Williams (1981) Prime graph components of finite groups J. Algebra 69 (2), 487-513
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