On the probability of being a ۲-Engel group

‎Let G be a finite group and d_۲(G) denotes the probability‎ ‎that [x,y,y]=۱ for randomly chosen elements x,y of G‎. ‎We‎ ‎will obtain lower and upper bounds for d_۲(G) in the case where‎ ‎the sets E_G(x)=\{y\in G:[y,x,x]=۱\} are subgroups of G for‎ ‎all x\in G‎. ‎Also the given examples illustrate that all the‎ ‎bounds are sharp‎.

Keywords:

Probability‎ , ‎۲-Engel condition‎ , ۳-metabelian