Hall Graph of a Finite Group


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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