OD-Characterization of certain four dimensional linear groups with related results concerning degree patterns

doi:10.1007/s11464-014-0430-2

Front. Math. China

2015, Vol. 10

Issue (1) : 1-31 https://doi.org/10.1007/s11464-014-0430-2

RESEARCH ARTICLE

OD-Characterization of certain four dimensional linear groups with related results concerning degree patterns

B. AKBARI,A. R. MOGHADDAMFAR(

)

Department of Mathematics, K. N. Toosi University of Technology, P. O. Box 16315-1618, Tehran, Iran Research Institute for Fundamental Sciences, Tabriz, Iran

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Abstract

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<?<p_k be all prime divisors of |G|. Then the degree pattern of G is defined as D(G) = (degG(p₁), degG(p₂), ? , degG(p_k)), 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).

Keywords Prime graph degree pattern simple group

Corresponding Author(s): A. R. MOGHADDAMFAR

Issue Date: 30 December 2014

Cite this article:

B. AKBARI,A. R. MOGHADDAMFAR. OD-Characterization of certain four dimensional linear groups with related results concerning degree patterns[J]. Front. Math. China, 2015, 10(1): 1-31.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-014-0430-2
https://academic.hep.com.cn/fmc/EN/Y2015/V10/I1/1

1	Akbari B, Moghaddamfar A R. Recognizing by order and degree pattern of some projective special linear groups. Internat J Algebra Comput, 2012, 22(6) (22 pages)
2	Akbari M, Moghaddamfar A R. Simple groups which are 2-fold OD-characterizable. Bull Malays Math Sci Soc, 2012, 35(1): 65-77
3	Akbari M, Moghaddamfar A R, Rahbariyan S. A characterization of some finite simple groups through their orders and degree patterns. Algebra Colloq, 2012, 19(3): 473-482 https://doi.org/10.1142/S1005386712000338
4	Brandl R, Shi W J. A characterization of finite simple groups with abelian Sylow 2-subgroups. Ric Mat, 1993, 42(1): 193-198
5	Carter R W. Simple Groups of Lie Type. Pure and Applied Mathematics, Vol 28. London-New York-Sydney: John Wiley and Sons, 1972
6	Conway J H, Curtis R T, Norton S P, Parker R A, Wilson R A. Atlas of Finite Groups. Oxford, Clarendon Press, 1985
7	Deng H, Shi W J. The characterization of Ree groups 2F4(q) by their element orders. J Algebra, 1999, 217(1): 180-187 https://doi.org/10.1006/jabr.1998.7808
8	Hoseini A A, Moghaddamfar A R. Recognizing alternating groups Ap+3 for certain primes p by their orders and degree patterns. Front Math China, 2010, 5(3): 541-553 https://doi.org/10.1007/s11464-010-0011-y
9	Khosravi B. Some characterizations of L9(2) related to its prime graph. Publ Math Debrecen, 2009, 75(3-4): 375-385
10	Kogani-Moghaddam R, Moghaddamfar A R. Groups with the same order and degree pattern. Sci China Math, 2012, 55(4): 701-720 https://doi.org/10.1007/s11425-011-4314-6
11	Kondratév A S. On prime graph components of finite simple groups. Mat Sb, 1989, 180(6): 787-797
12	Kondratév A S, Mazurov V D. Recognition of alternating groups of prime degree from their element orders. Sib Math J, 2000, 41(2): 294-302 https://doi.org/10.1007/BF02674599
13	Lucido M S, Moghaddamfar A R. Groups with complete prime graph connected components. J Group Theory, 2004, 7(3): 373-384 https://doi.org/10.1515/jgth.2004.013
14	Mazurov V D. Recognition of finite simple groups S4(q) by their element orders. Algebra Logic, 2002, 41(2): 93-110 https://doi.org/10.1023/A:1015356614025
15	Moghaddamfar A R. Recognizability of finite groups by order and degree pattern. In: Proceedings of the International Conference on Algebra 2010. 2010, 422-433
16	Moghaddamfar A R, Rahbariyan S. More on the OD-characterizability of a finite group. Algebra Colloq, 2011, 18(4): 663-674 https://doi.org/10.1142/S1005386711000514
17	Moghaddamfar A R, Rahbariyan S. OD-Characterization of some linear groups over binary field and their automorphism groups. Comm Algebra (to appear)
18	Moghaddamfar A R, Zokayi A R. Recognizing finite groups through order and degree pattern. Algebra Colloq, 2008, 15(3): 449-456 https://doi.org/10.1142/S1005386708000424
19	Moghaddamfar A R, Zokayi A R. OD-Characterization of alternating and symmetric groups of degrees 16 and 22. Front Math China, 2009, 4(4): 669-680 https://doi.org/10.1007/s11464-009-0037-1
20	Moghaddamfar A R, Zokayi A R. OD-Characterization of certain finite groups having connected prime graphs. Algebra Colloq, 2010, 17(1): 121-130 https://doi.org/10.1142/S1005386710000143
21	Moghaddamfar A R, Zokayi A R, Darafsheh M R. A characterization of finite simple groups by the degrees of vertices of their prime graphs. Algebra Colloq, 2005, 12(3): 431-442 https://doi.org/10.1142/S1005386705000398
22	Rose J S. A Course on Group Theory. Cambridge: Cambridge University Press, 1978
23	Shi W J. A characterization of Suzuki’s simple groups. Proc Amer Math Soc, 1992, 114(2): 589-591 https://doi.org/10.1090/S0002-9939-1992-1074758-0
24	Suzuki M. On the prime graph of a finite simple group—an application of the method of Feit-Thompson-Bender-Glauberman. In: Groups and Combinatorics—in Memory of Michio Suzuki. Adv Stud Pure Math, 32. Tokyo: Math Soc Japan, 2001, 41-207
25	Vasilev A V, Gorshkov I B. On the recognition of finite simple groups with a connected prime graph. Sib Math J, 2009, 50(2): 233-238 https://doi.org/10.1007/s11202-009-0027-2
26	Vasilev A V, Vdovin E P. An adjacency criterion in the prime graph of a finite simple group. Algebra Logic, 2005, 44(6): 381-406 https://doi.org/10.1007/s10469-005-0037-5
27	Vasilev A V, Vdovin E P. Cocliques of maximal size in the prime graph of a finite simple group. Algebra Logic, 2011, 50(4): 291-322 https://doi.org/10.1007/s10469-011-9143-8
28	Williams J S. Prime graph components of finite groups. J Algebra, 1981, 69(2): 487-513 https://doi.org/10.1016/0021-8693(81)90218-0
29	Yan Y X, Chen G Y. OD-characterization of alternating and symmetric groups of degree 106 and 112. In: Proceedings of the International Conference on Algebra 2010. 2010, 690-696
30	Yan Y X, Chen G Y, Wang L L. OD-characterization of the automorphism groups of O10(2). Indian J Pure Appl Math, 2012, 43(3): 183-195 https://doi.org/10.1007/s13226-012-0011-6
31	Zavarnitsine A V. Finite simple groups with narrow prime spectrum. Sib Elektron Mat Izv, 2009, 6: 1-12
32	Zhang L C, Shi W J. OD-Characterization of all simple groups whose orders are less than 108. Front Math China, 2008, 3(3): 461-474 https://doi.org/10.1007/s11464-008-0026-9
33	Zhang L C, Shi W J. OD-Characterization of almost simple groups related to L2(49). Arch Math (Brno), 2008, 44(3): 191-199
34	Zhang L C, Shi W J. OD-Characterization of simple K4-groups. Algebra Colloq, 2009, 16(2): 275-282 https://doi.org/10.1142/S1005386709000273
35	Zhang L C, Shi W J. OD-Characterization of almost simple groups related to U3(5). Acta Math Sin (Engl Ser), 2010, 26(1): 161-168 https://doi.org/10.1007/s10114-010-7613-x
36	Zhang L C, Shi W J. OD-characterization of almost simple groups related to U6(2). Acta Math Sci Ser B Engl Ed, 2011, 31(2): 441-450
37	Zhang L C, Shi W J. OD-Characterization of the projective special linear groups L2(q). Algebra Colloq, 2012, 19(3): 509-524 https://doi.org/10.1142/S1005386712000375
38	Zhang L C, Shi W J, Shao C G, Wang L L. OD-Characterization of the simple group L3(9). J Guangxi Univ Natur Sci Ed, 2009, 34(1): 120-122
39	Zhang L C, Shi W J, Wang L L, Shao C G. OD-Characterization of A16. J Suzhou Univ Natur Sci Ed, 2008, 24(2): 7-10
40	Zsigmondy K. Zur Theorie der Potenzreste. Monatsh Math Phys, 1892, 3(1): 265-284 https://doi.org/10.1007/BF01692444

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