On the probability of being a $2$-Engel group

Document Type: Research Paper

Authors

Ferdowsi University of Mashhad

Abstract

‎Let $G$ be a finite group and $d_2(G)$ denotes the probability‎ ‎that $[x,y,y]=1$ for randomly chosen elements $x,y$ of $G$‎. ‎We‎ ‎will obtain lower and upper bounds for $d_2(G)$ in the case where‎ ‎the sets $E_G(x)=\{y\in G:[y,x,x]=1\}$ are subgroups of $G$ for‎ ‎all $x\in G$‎. ‎Also the given examples illustrate that all the‎ ‎bounds are sharp‎.

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