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Summary of Chapter 15, Quotient Groups Theorem 1: If H is a normal subgroup of group G, then aH = Ha for every the left and right cosets of a normal subgroup are the same). a ∈ G . (In other words, Theorem 2: Let H be a normal subgroup of group G. If Ha = Hc and Hb = Hd , then H (ab) = H (cd ) . (Thus, coset multiplication is uniquely defined for normal subgroups.) Definition: Let H be a normal subgroup of group G. Then the symbol G H denotes the set of all cosets of H in G. So, G H = {Ha , Hb, Hc,...} if Ha, Hb, Hc, ... are cosets of H in G. Theorem 3: G H with coset multiplication is a group, called the quotient group of G by H. Theorem 4: G H is a homomorphic image of G. (In other words, there is a homomorphism, namely the natural homomorphism, f ( x ) = Hx , from G onto G H .) Theorem 5: Let G be a group and H a subgroup of G. Then (i) Ha = Hb if and only if ab −1 ∈ H . (ii) Ha = H if and only if a ∈ H . Example: If G is a group, a commutator of G is any element of G of the form aba −1b −1 where a , b ∈ G . (Note that aba −1b −1 = e if and only if ab = ba .) The set H consisting of all the commutators of G is a normal subgroup of G. In a sense, the more commutators that G has, the less abelian G is. The quotient group G H is abelian. By moving all commutators of G into a subgroup H, we “caused” the quotient group G H , a homomorphic image of G, to be abelian. This demonstrates how we can sometimes “tailor” a homomorphic image G H of G to have specific properties by wisely choosing a normal subgroup H of G.