# Thread: Proving a group is Abelian...

1. ## Proving a group is Abelian...

Can someone help me with this...TIA

Let G be a group which satisfies the following property:
Whenever x,y,z are elements of G such that xy=yz, then x=z.
Prove that G must be Abelian.

I don't even know how to start. TIA...

2. Originally Posted by ginafara
Can someone help me with this...TIA

Let G be a group which satisfies the following property:
Whenever x,y,z are elements of G such that xy=yz, then x=z.
Prove that G must be Abelian.

I don't even know how to start. TIA...
I have NO experience with Abstract Algebra whatsoever, but for some reason i felt like taking a stab at this.

According to Wikipedia, an Abelian group is a group (G,*) such that a*b = b*a for all a,b in G.

now we are told that $\displaystyle xy = yz \implies x = z$, for any $\displaystyle x,y,z$ in G

this means that we can write, $\displaystyle xy = yx$ or alternatively, $\displaystyle zy = yz$. Which is the exact definition of an Abelian Group.

don't take my word for it though, wait until the experts get here

3. I have no experience with A.A. either...I am strictly Applied... I will try and get it going... thanks

4. By associativity, we have $\displaystyle (ab)a=a(ba)$.
Now, take $\displaystyle x=ab$, $\displaystyle z=ba$ and $\displaystyle y=a$.
Thus, $\displaystyle xy=yz\Rightarrow x=z\Rightarrow ab=ba$.