
commutativity of a group
I'm trying to prove this problem but I'm not sure whether or not my answer is working. The problem states
"Let G be a group, with the following property: Whenever a, b, and c belong to G and ab=ca, then b=c. Prove that G is abelian."
I have solved for b and c using inverses, and got that b=a1(ca) and c=(ab)a1. The second half of the problem states that b=c. So I wanted to use replacement to show that the group is abelian. But I'm not sure if that would prove G is abelian for any arbitrary b and c or if its only commutative for this exact pair.

Hi
Perhaps it is better to start with two elements, let's say $\displaystyle a,b,$ in $\displaystyle G.$ You want to prove $\displaystyle ab=ba$.
Inverses are indeed often useful for group equations: take a look at $\displaystyle ab=ab(a^{1}a)$ using associativity and your hypothesis.

I'm sorry I don't think I understand. I could regroup by the associative property stating
ab=(aba1)a
and I previously found that c=(aba1) so by replacement ab=ca so therefore b=c. But I'm still confused about how to prove the group is abelian. I'm missing something.

Well, you have $\displaystyle ab=(aba^{1})a$
You can define $\displaystyle c:=aba^{1}$ if you want, so, by hypothesis, you have: $\displaystyle b=c,$ i.e. $\displaystyle b=aba^{1},$ and you can conclude.

yea i realized that a couple minutes after i had replied. (Headbang) thank you sooo much for the explanation!