If every left coset xH of a subgroup H in a group G is equal to a right coset Hy for some y in G, then H is a normal subgroup of G. A normal subgroup is defined as a subgroup that is invariant under conjugation by elements of the group, meaning for all g in G and h in H, the element gHg⁻¹ is still in H. The proof involves showing that the condition of equal cosets implies this invariance. Thus, the equality of left and right cosets is a sufficient condition for normality in group theory. Understanding this concept is crucial for further studies in abstract algebra.