Abstract
We show that a finite group satisfying the law $$[y,_nx]=[x,_ny]$$ ( $$n>1$$ ) is nilpotent and utilizing the results of Macdonalds on the structure of groups satisfying the law $$[y,x]=[x,y]$$ , we investigate groups satisfying both of the laws $$[y,x]=[x,y]$$ and $$[y,_nx]=[x,_ny]$$ 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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