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

Document Type : Research Paper

Authors

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.

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 $D_n(q)$.

Keywords

Main Subjects

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### History

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