Groups described by element numbers

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Abstract

Let $G$ be a finite group and $L_e(G)=\{x \in G \ | \ x^e=1\}$, where $e$ is a positive integer dividing $|G|$. How do bounds on $|L_e(G)|$ influence the structure of $G$ ? Meng and Shi [W. Meng and J. Shi, On an inverse problem of Frobenius' theorem, Arch. Math. (Basel) 96 (2011), 109--114] have answered this question for $|L_e(G)| \le 2e$. 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.
Lingua originale English 10 Forum Mathematicum in stampa Published - 2013

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