Recognition of alternating groups of prime degree from their element orders. (English. Russian original) Zbl 0956.20007
Sib. Math. J. 41, No. 2, 294-302 (2000); translation from Sib. Mat. Zh. 41, No. 2, 359-369 (2000).
Let \(G\) be a group. The set of the orders of the elements is denoted by \(\omega(G)\). The following theorem is proven: Let \(G\) be a finite group such that \(\omega(G)=\omega(A_r)\), where \(A_r\) is the alternating group of degree \(r\) and \(r>3\) is a prime. Then \(G\cong A_r\).
Recently, A. V. Zavarnitsyn generalized this result and proved an analogous theorem for the alternating group \(A_r\), where either \(r=p+1\) or \(r=p+2\) and \(p>7\) is a prime.
Recently, A. V. Zavarnitsyn generalized this result and proved an analogous theorem for the alternating group \(A_r\), where either \(r=p+1\) or \(r=p+2\) and \(p>7\) is a prime.
Reviewer: E.P.Vdovin (Novosibirsk)
MSC:
| 20D06 | Simple groups: alternating groups and groups of Lie type |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 20D05 | Finite simple groups and their classification |
Keywords:
prime graphs; Gruenberg-Kegel graphs; finite simple groups; alternating groups; element ordersReferences:
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