Group theory question

1. Dec 20, 2007

quasar987

1. The problem statement, all variables and given/known data
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 $$c_{b_2}(a_1)=a_1$$

3. 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 $$a_1a=a_2a'$$??

2. Dec 20, 2007

morphism

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