Is there a way to find a solution to the equation ax=b in the group G?

[SNOOTCHIEBOOCHEE]

Homework Statement

let a,b $$\in$$G show that ax=b has a unique solution in G

The Attempt at a Solution

i know what needs to be done, i just don't know how to do it.

Want to prove:

1. There is a solution
2. solution is unique

to prove uniquness of a soltuion just suppose you have a different solution x' and show that x'=x


to show that there is a solution (im not sure this part is right, cause it seems too simple) simply multiply (left) by $$a^{-1}$$

that is $$a^{-1}$$*ax=$$a^{-1}$$b

we can do this because a is in the group, which implies $$a^{-1}$$ is in the group

THis is where i am stuck... i really don't know how to proceed. Have i even done the first part right?

any help appreciated

 

[morphism]

So you showed that there is a solution, namely $x=a^{-1}b$. To show that x is unique, use the fact that in a group, inverses are unique.