On the probability of being a 2-Engel group

Abstract

‎Let $G$ be a finite group and $d_2(G)$ denotes the probability‎that $[x,y,y]=1$ for randomly chosen elements $x,y$ of $G$‎. ‎We‎will obtain lower and upper bounds for $d_2(G)$ in the case where‎the sets $E_G(x)={yin G:[y,x,x]=1}$ are subgroups of $G$ for‎all $xin G$‎. ‎Also the given examples illustrate that all the‎bounds are sharp.

Keywords

• Probability‎
• ‎2-Engel condition‎
• 3-metabelian