Hall Graph of a Finite Group

doi:10.1285/i15900932v39n2p25

Hall Graph of a Finite Group

B. Akbari

Abstract

The Hall graph of a finite group $G$ is a simple graph whose vertex set is $\pi(G)$ , the set of all prime divisors of its order, and two distinct primes $p$ and $q$ are joined by an edge if $G$ has at least one Hall $\{p, q\}$ -subgroup. For all primes $p_1<\cdots<p_k$ of $\pi(G)$ , we call the $k$ -tuple ${\rm D}_{\rm H}(G)=(d_{\rm H}(p_1), \ldots, d_{\rm H}(p_k))$ , the degree pattern of Hall graph of $G$ , where $d_{\rm H}(p)$ signifies the degree of vertex $p$ . This paper provides some properties of Hall graph. It also gives a characterization for some finite simple groups via order and degree pattern of Hall graph.

DOI Code: 10.1285/i15900932v39n2p25

Keywords: Hall graph; degree pattern of Hall graph; Hall subgroup; simple group

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This work is licensed under a Creative Commons Attribuzione - Non commerciale - Non opere derivate 3.0 Italia License.

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Note di Matematica

Hall Graph of a Finite Group

Abstract