Proof of the inverse of an inverse

[fk378]

Homework Statement

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)

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.

 

[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).

 

[fk378]

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

 

[Dick]

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

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

 

[statdad]

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