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.