Proving group commutativity

I have a homework problem that 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 started with the hypothesis ab=ca and solved for b and c using inverses. I found b=(a-1)ca and c=ab(a-1). Because the hypothesis says b=c I set them equal. (a-1)ca=ab(a-1). But I'm having trouble getting anywhere useful after that. Hints or suggestions if I'm on the right track???
 
Suppose G is Abelian and it doesn't have that property. Show a contradiction that G must not be Abelian.

Can you show a contradiction?
 
So I should try assuming its not abelian and try proving that if ab=ca, then b equals c??
 
So I should try assuming its not abelian and try proving that if ab=ca, then b equals c??
Suppose G is Abelian and it doesn't have that property.

I already wrote the first line for you.
 
402
1
Start with [tex]aba^{-1}=c[/tex] (c is just some element of the group). Now, if you right-multiply by a, and apply the hypothesis, what do you conclude?
 

Related Threads for: Proving group commutativity

Replies
6
Views
773
Replies
3
Views
1K
Replies
3
Views
1K
Replies
6
Views
2K
Replies
1
Views
2K
Replies
5
Views
822

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top