1. The problem statement, all variables and given/known data I think I've got this one about figured out, I just wanted someone to check it over. (For this problem, (a-1) is a inverse, (b-1) b inverse, etc.) "Let G be a group, H a subgroup of G. Then, H is normal in G iff every left coset of H is equal to some right coset of H" 2. Relevant equations H normal in G implies aH(a-1) = H for all a in G 3. The attempt at a solution Let a,b be elts of G, and H a normal sbgp of G. (=>): By normal condition, aH(a-1) = H for all a in G -> aH = Ha (right multiply by a) H normal in G implies the left coset of H by a coincides with the right coset of H by a. Implies that if H is normal, every left coset of H is equal to some right coset of H. (<=): Let the left coset of H by a be equal to some right coset of H by b. So, aH = Hb -> aH(b-1) = H (right multiply by (b-1) ) H = aH(b-1) implies (by taking inverses) that H = bH(a-1) Finally, H = HH = (aH(b-1))(bH(a-1)) = aH((b-1)b)H(a-1) = aH(e)H(a-1) = aHH(a-1) = aH(a-1) Then we have H = aH(a-1) Similar argument shows that H = bH(b-1) So, H is normal. In conclusion, if every left coset of H is equal to some right coset of H, then H is normal in G. I think that handles both directions (<=) & (=>). To me, this seems like a pretty pertinent question regarding normal sbgps so I'd like to make sure I hit it on the head. Thanks!