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Groups described by element numbers. (English) Zbl 1434.20008

Summary: Let \(G\) be a finite group and \(L_e(G)=\{ x \in G\mid x^e=1\}\), where \(e\) is a positive integer dividing \(|G|\). How do bounds on \(|L_e(G)|\) influence the structure of \(G\)? W. Meng and J. Shi [Arch. Math. 96, No. 2, 109–114 (2011; Zbl 1225.20023)] have answered this question for \(|L_e(G)|\leq 2e\). We generalize their contributions, considering the inequality \(|L_e(G)| \le e^2\) and finding a new class of groups of whose we study the structural properties.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D15 Finite nilpotent groups, \(p\)-groups

Citations:

Zbl 1225.20023
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