# Group theory question

1. Jan 30, 2008

### ehrenfest

1. The problem statement, all variables and given/known data
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

2. Relevant equations

3. 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

Last edited: Jan 30, 2008
2. Jan 30, 2008

### NateTG

Yes, it's true.

3. Jan 30, 2008

### ircdan

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.

4. Jan 30, 2008

### 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

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