Group Theory Question: Is (Left) Multiplication by g an Isomorphism in G?

[ehrenfest]

Homework Statement

true or false:
If G is a group and g is in G. Then (left) multiplication by g is an isomorphism from G to G

Homework Equations

The Attempt at a Solution

I am pretty sure it is true since ax=b always has a solution if a and b are in group. But can someone just confirm this?

EDIT: sorry, I don't mean isomorphism, I mean bijection

 

[NateTG]

Yes, it's true.

 

[ircdan]

Homework Statement

I am pretty sure it is true since ax=b always has a solution if a and b are in group. But can someone just confirm this?

EDIT: sorry, I don't mean isomorphism, I mean bijection

it's true as nate said, but your reason doesn't tell the whole story(it just gives surjectivity). Try to do it directly, for a fixed g in G, define phi:G->G by phi(x) = gx. Now show it's a bijection.

 

[jacobrhcp]

gx is in g, and if g1x=g2x then g2inv g1 x = x so g1=g2 so all elements are different, so you can just make couple in your head from every g to every gx.

I passed group theory this monday =D