## Abstract

Let *G* be a finite group and ${L}_{e}\left(G\right)=\{x\in G\mid {x}^{e}=1\}$, where *e* is a positive integer dividing $\left|G\right|$. How do bounds on $\left|{L}_{e}\left(G\right)\right|$ influence the structure of *G*?
Meng and Shi [Arch. Math. (Basel) 96 (2011), 109–114] have answered this question for $\left|{L}_{e}\left(G\right)\right|\le 2e$. We generalize their contributions, considering the inequality $\left|{L}_{e}\left(G\right)\right|\le {e}^{2}$ and finding a new class of groups of whose we study the structural properties.

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