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A new characterization of the sporadic simple groups. (English) Zbl 0657.20017
Group theory, Proc. Conf., Singapore 1987, 531-540 (1989).
[For the entire collection see Zbl 0652.00004.]
Let \(H\) be a finite, sporadic simple group. The author shows, that if \(G\) is a group with \(|G|=|H|\) and \(G\) has the same set of orders of its elements as \(H\), one has \(G\simeq H\). This paper is a successor of similar investigations of the author [Adv. Math., Beijing 16, 397-401 (1987; Zbl 0643.20008)]; J. Southwest-China Teachers Univ., Ser. B 1, 36-40 (1984); ibid. 4, 1-8 (1987)]. The two main ingredients of the proof are the classification of finite simple groups and a paper of J. S. Williams [J. Algebra 69, 487-513 (1981; Zbl 0471.20013)]. The second result together with the assumptions show that \(G\) has precisely one simple, nonabelian composition factor with some additional information on the prime divisors of its order. From here on just the knowledge of the orders of finite simple groups and the orders of some of their elements together with numerical considerations are enough to yield the above result.
Reviewer: U.Dempwolff

20D08 Simple groups: sporadic groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups