Recognizing alternating groups A p +3 for certain primes p by their orders and degree patterns

doi:10.1007/s11464-010-0011-y

Front. Math. China

2010, Vol. 5

Issue (3) : 541-553 https://doi.org/10.1007/s11464-010-0011-y

Research articles

Recognizing alternating groups A p +3 for certain primes p by their orders and degree patterns

A. A. HOSEINI¹,A. R. MOGHADDAMFAR²,

1.Department of Mathematics, K. N. Toosi University of Technology, P. O. Box 16315-1618, Tehran, Iran; 2.Department of Mathematics, K. N. Toosi University of Technology, P. O. Box 16315-1618, Tehran, Iran；School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran;

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Abstract The degree pattern of a finite group M has been introduced by A. R. Moghaddamfar et al. [Algebra Colloquium, 2005, 12(3): 431―442]. A group M is called k-fold OD-characterizable if there exist exactly k nonisomorphic finite groups having the same order and degree pattern as M. In particular, a 1-fold OD-characterizable group is simply called OD-characterizable. In this article, we will show that the alternating groups A_p+3 for p = 23, 31, 37, 43 and 47 are OD-characterizable. Moreover, we show that the automorphism groups of these groups are 3-fold OD-characterizable. It is worth mentioning that the prime graphs associated with all these groups are connected.

Keywords OD-characterization of finite group degree pattern prime graph alternating and symmetric groups

Issue Date: 05 September 2010

Cite this article:

A. A. HOSEINI,A. R. MOGHADDAMFAR. Recognizing alternating groups A p +3 for certain primes p by their orders and degree patterns[J]. Front. Math. China, 2010, 5(3): 541-553.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-010-0011-y
https://academic.hep.com.cn/fmc/EN/Y2010/V5/I3/541

	Aschbacher M. FiniteGroup Theory. Cambridge: Cambridge University Press, 1986
	Carter R W. Simple Groups of Lie type. Pure and AppliedMathematics, Vol 28. London-NewYork-Sydney: John Wiley and Sons, 1972
	Conway J H, Curtis R T, Norton S P, Parker R A, Wilson R A. Atlas of Finite Groups. Oxford: ClarendonPress, 1985
	Darafsheh M R, Moghaddamfar A R, Zokayi A R. A characterization of some symmetric and linear groups. Algebras Groups Geom, 2001, 18(3): 281―299
	Kleidman P, Liebeck M. The Subgroup Structure ofthe Finite Classical Groups. London MathematicalSociety Lecture Note Series, 129. Cambridge: Cambridge UniversityPress, 1990
	Kondratév A S. On prime graph components of finite simple groups. Math Sb, 1989, 180(6): 787―797
	Lucido M S. Prime graph components of finite almost simple groups. Rend Sem Mat Univ Padova, 1999, 102: 1―22
	Mazurov V D. On the set of orders of elements of a finite group. Algebra and Logic, 1994, 33(1): 49―55 doi: 10.1007/BF00739417
	Mazurov V D. Characterizations of finite groups by sets of the orders of theirelements. Algebra and Logic, 1997, 36(1): 23―32 doi: 10.1007/BF02671951
	Moghaddamfar A R, Zokayi A R. Recognizing finite groupsthrough order and degree pattern. AlgebraColloquium, 2008, 15(3): 449―456
	Moghaddamfar A R, Zokayi A R. OD-Characterization of alternatingand symmetric groups of degrees 16 and 22. Front Math China, 2009, 4(4): 669―680 doi: 10.1007/s11464-009-0037-1
	Moghaddamfar A R, Zokayi A R. OD-Characterization of certainfinite groups having connected prime graphs. Algebra Colloquium, 2010, 17(1): 121―130
	Moghaddamfar A R, Zokayi A R, Darafsheh M R. A characterization of finite simple groups by the degreesof vertices of their prime graphs. AlgebraColloquium, 2005, 12(3): 431―442
	Williams J S. Prime graph components of finite groups. J Algebra, 1981, 69(2): 487―513 doi: 10.1016/0021-8693(81)90218-0
	Zavarnitsin A V. Recognition of alternating groups of degrees r + 1 and r + 2 forprime r and the group of degree 16 by their element order sets. Algebra and Logic, 2000, 39(6): 370―377 doi: 10.1023/A:1010218618414
	Zavarnitsin A V, Mazurov V D. Element orders in coveringof symmetric and alternating groups. Algebraand Logic, 1999, 38(3): 159―170 doi: 10.1007/BF02671740
	Zavarnitsine A V. Finite simple groups with narrow prime spectrum. Siberian Electronic Mathematical Reports, 2009, 6: 112
	Zhang L C, Liu X. OD-Characterization of theprojective general linear groups PGL(2, q) by their orders and degree patterns. International Journal of Algebra and Computation, 2009, 19(7): 873―889 doi: 10.1142/S0218196709005433
	Zhang L C, Shi W J. OD-Characterization of allsimple groups whose orders are less than 10⁸. Front Math China, 2008, 3(3): 461―474 doi: 10.1007/s11464-008-0026-9
	Zhang L C, Shi W J. OD-Characterization of almostsimple groups related to L₂(49). Arch Math (Brno), 2008, 44(3): 191―199
	Zhang L C, Shi W J. OD-Characterization of simple K₄-groups. Algebra Colloquium, 2009, 16(2): 275―282
	Zhang L C, Shi W J. OD-Characterization of almostsimple groups related to U₃(5). Acta Mathematica Sinica(English Series), 2010, 26(1): 161―168 doi: 10.1007/s10114-010-7613-x
	Zhang L C, Shi W J. OD-Characterization of theprojective special linear group L₂(q). Algebra Colloquium (to appear)
	Zhang L C, Shi W J, Wang L L, Shao C G. OD-Characterizationof A₁₆. Journal of Suzhou University (Natural Science Edition), 2008, 24(2): 7―10

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