Khosravi, B., Babai, A. (2016). Simple groups with the same prime graph as $D_n(q)$. Bulletin of the Iranian Mathematical Society, 42(6), 1403-1427.

B. Khosravi; A. Babai. "Simple groups with the same prime graph as $D_n(q)$". Bulletin of the Iranian Mathematical Society, 42, 6, 2016, 1403-1427.

Khosravi, B., Babai, A. (2016). 'Simple groups with the same prime graph as $D_n(q)$', Bulletin of the Iranian Mathematical Society, 42(6), pp. 1403-1427.

Khosravi, B., Babai, A. Simple groups with the same prime graph as $D_n(q)$. Bulletin of the Iranian Mathematical Society, 2016; 42(6): 1403-1427.

Simple groups with the same prime graph as $D_n(q)$

^{1}Department of Pure Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, 424, Hafez Ave., Tehran 15914, Iran.

^{2}Department of Mathematics, University of Qom, P.O. Box 37185-3766, Qom, Iran.

Receive Date: 23 April 2014,
Revise Date: 13 September 2015,
Accept Date: 14 September 2015

Abstract

Vasil'ev posed Problem 16.26 in [The Kourovka Notebook: Unsolved Problems in Group Theory, 16th ed., Sobolev Inst. Math., Novosibirsk (2006).] as follows: Does there exist a positive integer $k$ such that there are no $k$ pairwise nonisomorphic nonabelian finite simple groups with the same graphs of primes? Conjecture: $k = 5$.

In [Zvezdina, On nonabelian simple groups having the same prime graph as an alternating group, Siberian Math. J., 2013] the above conjecture is positively answered when one of these pairwise nonisomorphic groups is an alternating group.

In this paper, we continue this work and determine all nonabelian simple groups, which have the same prime graph as the nonabelian simple group $D_n(q)$.

A. Babai, B. Khosravi and N. Hasani, Quasirecognition by prime graph of 2Dp(3) where p = 2n + 1 ≥5 is a prime, Bull. Malays. Math. Sci. Soc. (2) 32 (2009), no. 3, 343--350.

A. Babai and B. Khosravi, Recognition by prime graph of 2D2m+1(3), Sib. Math. J. 52 (2011), no. 5, 788--795.

A. Babai and B. Khosravi, Quasirecognition by prime graph of 2Dn(3α), where n =4m + 1 ≥21 and αis odd, Math. Notes, to appear.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford Univ. Press, Oxford, 1985.

P. Crescenzo, A diophantine equation which arises in the theory of finite groups, Adv. Math. 17 (1975), no. 1, 25--29.

M. Hagie, The prime graph of a sporadic simple group, Comm. Algebra 31 (2003), no. 9, 4405--4424.

A. Khosravi and B. Khosravi, A new characterization of some alternating and symmetric groups (II), Houston J. Math. 30 (2004) 465--478.

A. Khosravi and B. Khosravi, Quasirecognition by prime graph of the simple group 2G2(q), Sib. Math. J. 48 (2007), no. 3, 570--577.

V. D. Mazurov and E. I. Khukhro (eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, Sobolev Inst. Math. Novosibirsk, 16th edition, 2006.

W. Sierpinski, Elementary Theory of Numbers, Monografie Matematyczne, 42, Panstwowe Wydawnictwo Naukowe, Warsaw, 1964.

A. V. Vasil'ev and E. P. Vdovin, An adjacency criterion in the prime graph of a finite simple group, Algebra Logic 44 (2005), no. 6, 381--405.

A. V. Vasil'ev and E. P. Vdovin, Cocliques of maximal size in the prime graph of a finite simple group, Algebra Logic 50 (2011), no. 4, 291--322.

A. V. Vasil'ev and M. A. Grechkoseeva, On the recognition of the finite simple orthogonal groups of dimension 2m, 2m + 1 and 2m + 2 over a field of characteristic 2, Sib. Math.J. 45 (2004), no. 3, 420--431.

A. V. Zavarnitsin, On the recognition of finite groups by the prime graph, Algebra Logic 43 (2006), no. 4, 220--231.

K. Zsigmondy, Zur theorie der potenzreste, Monatsh. Math. Phys. 3 (1892), no. 1, 265--284.

M. A. Zvezdina, On nonabelian simple groups having the same prime graph as an alternating group, Sib. Math. J. 54 (2013), no. 1, 47--55.