Acta mathematica scientia,Series A ›› 2013, Vol. 33 ›› Issue (6): 1169-1177.

• Articles • Previous Articles     Next Articles

New Characterizations of L15(2)

 ZHANG Liang-Cai*, ZHANG Miao, NIE Wen-Min   

  1. College of Mathematics and Statistics, Chongqing University, Chongqing Shapingba |401331
  • Received:2012-07-13 Revised:2013-02-07 Online:2013-12-25 Published:2013-12-25
  • Contact: ZHANG Liang-Cai, E-mail:zlc213@163.com; 1073165848@qq.com; niewenmin1129@163.com
  • Supported by:

    国家自然科学基金(11271301, 11171364, 10871032)和国家自然科学青年基金项目(11001226)资助

Abstract:

If G is a finite group, we define its prime graph Γ(G) as follows. The vertices of Γ(G)  are the primes dividing the
order of G and two distinct vertices p, q are joined by an edge, denoted by p~q, if and only if there is an element in G of order pq (see [7, 15]). Assume |G|=p1α1p2α2pkαk with primes p1<p2<…<pk and natural numbers αi. For pπ(G)={p1, p2, … , pk, define deg(p):=|{qπ(G)|q~p}|, which is called the degree of p. We also define D(G):=(deg(p1), deg(p2), …, deg(pk)), which is called the degree pattern of the group G. We say a group G is t-fold OD-characterizable if there exist exactly t non-isomorphic finite groups M such that |M|=|G| and D(M)=D(G) (see [11]). In particular, a 1-fold OD-characterizable  group is simply called an OD-characterizable  group. In the present paper, we  prove  that the projective special linear group L15(2) is  OD-characterizable by a newly introduced lemma  to deal with its connected prime graph. As a consequence of this result, we obtain that  L15(2) is OG-characterizable.

Key words: Finite simple group, Prime graph, Degree of a vertex, Degree pattern

CLC Number: 

  • 20D05
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