Why Does Conjugation in Abelian Groups Imply Triviality for Normal Subgroups?

[quasar987]

Homework Statement

G is abelian, A is normal in G, B is a subgroup, a1, a2 in A, b1,b2 in B, c_g denotes the congugation by g automorphism. why must

$$a_1c_{b_1}(a_2) = a_2c_{b_2}(a_1)$$

imply that $$c_{b_1}(a_2)=a_2$$ and $$c_{b_2}(a_1)=a_1$$

The Attempt at a Solution

In other words, why couldn't there exists a, a' in A such that $$c_{b_1}(a_2)=a$$ and $$c_{b_2}(a_1)=a'$$ and $$a_1a=a_2a'$$??

 

[morphism]

Conjugation is trivial in an abelian group: ghg^-1 = gg^-1h = h.