ON THE PROBABILITY OF BEING A 2-ENGEL GROUP

Let G be a finite group and d2 (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 d2 (G) in the case where the sets EG (x) = {y is an element of G : [y, x, x] = 1} are subgroups of G for all x is an element of G. Also the given examples illustrate that all the bounds are sharp.