Abstract In this talk, It is shown that a finite group G satisfying [y,n x] = [x,n y] ( for all x, y 2 G and n > 1) is nilpotent and that if G is a group satisfying [y, x] = [x, y], then [\\\\\\\\gamma_3(G), \\\\\\\\gamma_2(G) ] = [ \\\\\\\\gamma_2(G), \\\\\\\\gamma_2(G),G] = 1. Also, we investigate groups satisfying both [y, x] = [x, y] and [y,n x] = [x,n y] for small n. Our results can be applied to obtain special commutators, which can be expressed as the product of commutators squares.

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