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) = fy 2 G : [y; x; x] = 1g are subgroups of G for all x 2 G. Also the given examples illustrate that all the bounds are sharp.