# Proof of the inverse of an inverse

1. Sep 13, 2008

### fk378

1. The problem statement, all variables and given/known data
In any group, verify directly from the axioms that
(a) inverse of the inverse of x= x
(b) (xy)^inverse = (inverse y)(inverse x) for all x,y in G. (note the reversal here)

3. The attempt at a solution
(a) I tried to use the axiom that xe=x=ex but I don't know where to go from there.
(b) I don't know how to start it.

2. Sep 13, 2008

### Dick

If b is the inverse of a then ab=ba=e. If a is the inverse of b then ba=ab=e. They are the SAME THING. Think of what that means if a=x and b=x^(-1).

3. Sep 13, 2008

### fk378

So my proof should conclude with noticing that x is the inverse of x-inverse?

4. Sep 13, 2008

### Dick

Well, yes. It is, isn't it?

5. Sep 13, 2008

For the second question - what happens if you multiply $$xy$$ with the object you need to show is its inverse?