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On decomposability of finite groups. (English) Zbl 1488.20061

Summary: Let \(G\) be a finite group. A normal subgroup \(N\) of \(G\) is a union of several \(G\)-conjugacy classes, and it is called \(n\)-decomposable in \(G\) if it is a union of \(n\) distinct \(G\)-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5.

MSC:

20E45 Conjugacy classes for groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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