Characterization of some alternating groups by order and largest element order

Document Type : Original Article

Authors

Department of Mathematics, North Tehran Branch, Islamic Azad University, Tehran, Iran

Abstract

The prime graph (or Gruenberg-Kegel graph) of a finite group is a well-known graph. In this paper, first, we investigate the structure of the finite groups with a non-complete prime graph. Then as an application, we prove that every alternating group An, where n ≤ 31 is determined by its order and its largest element order. Also, we show that A32 is not characterizable by order and the largest element order.

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Main Subjects


[1] M. Bibak, G. Rezaeezadeh, and E. Esmaeilzade, A new characterization of a simple group G2(q) where q 6 11, J. Algebr. Syst., 8 (2020), pp. 103–111, 10 (Persian pp.).
[2] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ATLAS of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups, With computational assistance from J. G. Thackray.
[3] B. Ebrahimzadeh, A. Iranmanesh, A. Tehranian, and H. Parvizi Mosaed, A characterization of the Suzuki groups by order and the largest elements order, J. Sci. Islam. Repub. Iran, 27 (2016), pp. 353–355.
[4] B. Ebrahimzadeh, R. Mohammadyari, and M. Sadeghi, A new characterization of the simple groups C4(q), by its order and the largest order of elements, Acta Comment. Univ. Tartu. Math., 23 (2019), pp. 283– 290.
[5] I. Gorshkov and A. Staroletov, On groups having the prime graph as alternating and symmetric groups, Comm. Algebra, 47 (2019), pp. 3905–3914.
[6] I. B. Gorshkov and N. V. Maslova, The group J4 × J4 is recognizable by spectrum, J. Algebra Appl., 20 (2021), pp. Paper No. 2150061, 14.
[7] S. Guest, J. Morris, C. E. Praeger, and P. Spiga, On the maximum orders of elements of finite almost simple groups and primitive permutation groups, Trans. Amer. Math. Soc., 367 (2015), pp. 7665–7694.
[8] L.-G. He and G.-Y. Chen, A new characterization of L2(q) where q = p n < 125, Ital. J. Pure Appl. Math., (2011), pp. 127–136.
[9] L.-G. He, G.-Y. Chen, and H.-J. Xu, A new characterization of sporadic simple groups, Ital. J. Pure Appl. Math., (2013), pp. 373–392.
[10] I. M. Isaacs, Finite group theory, vol. 92 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2008.
[11] Q. H. Jiang, C. G. Shao, W. J. Shi, and Q. L. Zhang, A new characterization of L2(q) by the largest element orders, Bull. Iranian Math. Soc., 43 (2017), pp. 1143–1151.
[12] W. M. Kantor and A. Seress, Large element orders and the characteristic of Lie-type simple groups, J. Algebra, 322 (2009), pp. 802–832.
[13] J. Li, D. Yu, G. Chen, and W. Shi, A characterization of simple K4-groups of type L2(q) and their automorphism groups, Bull. Iranian Math. Soc., 43 (2017), pp. 501–514.
[14] J.-P. Massias, J.-L. Nicolas, and G. Robin, Effective bounds for the maximal order of an element in the symmetric group, Math. Comp., 53 (1989), pp. 665–678.
[15] A. V. Vasil’ev, On connection between the structure of a finite group and the properties of its prime graph, Siberian Mathematical Journal, 46 (2005), pp. 396–404.
[16] A. V. Vasil’ev, M. A. Grechkoseeva, and V. D. Mazurov, Characterization of the finite simple groups by spectrum and order, Algebra and Logic, 48 (2009), pp. 385–409.
[17] H. Xu, G. Chen, and Y. Yan, A new characterization of simple K3-groups by their orders and large degrees of their irreducible characters, Comm. Algebra, 42 (2014), pp. 5374–5380.
[18] A. V. Zavarnitsin and V. D. Mazurov, Element orders in coverings of symmetric and alternating groups, Algebra Log., 38 (1999), pp. 296–315, 378.
[19] A. V. Zavarnitsine, Finite simple groups with narrow prime spectrum, Sib. Elektron. Mat. Izv., 6 (2009), ` pp. 1–12.