Let G be a group and let $$x,y,z \in G$$. Assume that xyz=1. Does it follow that yzx=1?

Question

anonymous · Answer

I believe that since vector multiplication is commulative and that x,y,z are members of G, that both equations are the same. I could be totally wrong though.

kinggeorge · Answer

No, but $$z^{-1}y^{-1}x^{-1} = 1$$

jim_thompson5910 · Answer

but group commutativity is not guaranteed (G is a group, not a vector space), so we can't say that yzx=1 is true

anonymous · Answer

so if they specifically said that G is a commutative group then it would be true yes?

kinggeorge · Answer

If G is abelian (commutative), then it would be true.

zarkon · Answer

$$xyz=1$$
$$x^{-1}xyz=x^{-1}1=x^{-1}$$
$$yz=x^{-1}$$
$$yzx=1$$

jim_thompson5910 · Answer

yes, commutative being the key word there @91rt

jim_thompson5910 · Answer

oh didn't see zarkons work there, he does have a point

kinggeorge · Answer

Zarkon seems to be correct. That's rather interesting.

anonymous · Answer

thanks everyone!

anonymous · Answer

if 
$$xyz=1$$ then 
$$z=(xy)^{-1}=y^{-1}x^{-1}$$ and so 
$$yzx=yy^{-1}x^{-1}x=1$$
just another way of looking at it