in this article, we show the existence of certain exact sequences with respect to two homology theories, called d-homology and extended d-homology. we present sufficient conditions for the existence of long exact extended d- homology sequence. also we give some illustrative examples.

Let nand k be positive integers. We say that a group G satisfies the condition £(n) (respectively, £dn)) if and only if any set with n + 1 elements of G contains two distinct elements x, y such that [x,t y] = 1 for some positive integer t = t(x, y) (respectively, [X,k y] = 1). Here we study certain groups satisfying these conditions. We prove that if G is a finite group satisfying the condition...