Mazurov, V. D. Recognition of finite nonsimple groups by the set of orders of their elements. (English. Russian original) Zbl 0880.20017 Algebra Logika 36, No. 3, 304-322 (1997); translation in Algebra Logic 36, No. 3, 182-192 (1997). The article continues the study of Ya. G. Berkovich [Izv. Sev.-Kavk. Nauchn. Tsentr. Vyssh. Shk., Estestv. Nauki 1981, No. 1(33) 6-9 (1981; Zbl 0492.20005)], W. Shi [Proc. Am. Math. Soc. 114, 589-591 (1992; Zbl 0747.20007) and Algebra Colloq. 1, 159-166 (1994; Zbl 0799.20019)], Ch. E. Praeger and W. Shi [Commun. Algebra 22, 1507-1530 (1994; Zbl 0802.20015)], and R. Brandl and W. Shi [Ric. Mat. 42, 193-198 (1993; Zbl 0822.20012)] on recognition of finite groups by the set of orders of their elements. It is proven in the article that there exists a finite group \(G\) that contains a minimal normal subgroup \(N\) with an element of order equal to the period of \(N\), such that the group \(G\) is determined by the set of orders of its elements up to isomorphism. It is also proven that there exists a finite group that is recognized by the set of orders of its elements and decomposed into the direct product of two nontrivial subgroups. All the results do not use the classification of finite simple groups. Reviewer: E.P.Vdovin (Novosibirsk) Cited in 1 ReviewCited in 8 Documents MSC: 20D60 Arithmetic and combinatorial problems involving abstract finite groups Keywords:finite groups; orders of elements Citations:Zbl 0745.20022; Zbl 0747.20007; Zbl 0799.20019; Zbl 0802.20015; Zbl 0822.20012; Zbl 0492.20005 PDFBibTeX XMLCite \textit{V. D. Mazurov}, Algebra Logika 36, No. 3, 304--322 (1997; Zbl 0880.20017); translation in Algebra Logic 36, No. 3, 182--192 (1997) Full Text: EuDML