Heineken, Hermann; Russo, Francesco G. Groups described by element numbers. (English) Zbl 1434.20008 Forum Math. 27, No. 4, 1961-1977 (2015). 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. Cited in 1 ReviewCited in 3 Documents 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 Keywords:metacyclic groups; Frobenius’ theorem; \(p\)-groups; Sylow subgroups Citations:Zbl 1225.20023 PDFBibTeX XMLCite \textit{H. Heineken} and \textit{F. G. Russo}, Forum Math. 27, No. 4, 1961--1977 (2015; Zbl 1434.20008) Full Text: DOI