Frontiers of Mathematics in China >
OD-Characterization of certain four dimensional linear groups with related results concerning degree patterns
Received date: 31 Jan 2013
Accepted date: 12 Sep 2014
Published date: 30 Dec 2014
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The prime graph of a finite group G, which is denoted by GK(G), is a simple graph whose vertex set is comprised of the prime divisors of |G| and two distinct prime divisors p and q are joined by an edge if and only if there exists an element of order pq in G. Let p1<p2<<pk be all prime divisors of |G|. Then the degree pattern of G is defined as D(G) = (degG(p1), degG(p2), , degG(pk)), where degG(p) signifies the degree of the vertex p in GK(G). A finite group H is said to be OD-characterizable if G H for every finite group G such that |G| = |H| and D(G) = D(H). The purpose of this article is threefold. First, it finds sharp upper and lower bounds on (G), the sum of degrees of all vertices in GK(G), for any finite group G (Theorem 2.1). Second, it provides the degree of vertices 2 and the characteristic p of the base field of any finite simple group of Lie type in their prime graphs (Propositions 3.1-3.7). Third, it proves the linear groups L4(q), q = 19, 23, 27, 29, 31, 32, and 37, are OD-characterizable (Theorem 4.2).
Key words: Prime graph; degree pattern; simple group
B. AKBARI , A. R. MOGHADDAMFAR . OD-Characterization of certain four dimensional linear groups with related results concerning degree patterns[J]. Frontiers of Mathematics in China, 2015 , 10(1) : 1 -31 . DOI: 10.1007/s11464-014-0430-2
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