Left and right identity 
Possible Duplicate:
Prove if an element of a monoid has an inverse, that inverse is unique 

How to show that the left inverse x' is also a right inverse, i.e, x * x' = e?
Also, how can we show that the left identity element e is a right identity element also? 
Thanks
 A: The idea for these uniqueness arguments is often this: take your identities and try to get them mixed up with each other. Assuming that you are working with groups, suppose that we have $x, y, z$ in a group such that $yx = xz = e$. The products $(yx)z$ and $y(xz)$ are equal, because the group operation is associative. Evaluate these as written and see what happens. The story for left/right identities is even simpler: if I have two elements in a group, what's the obvious thing to do with them?
A: The argument for identities is very simple: Assume we have a group G with a left identity g and a right identity h.Then strictly by definition of the identity:
      g = gh = h.
 So g=h. Q.E.D.
The argument for inverses is a little more involved,but the basic idea is given for inverses below by Dylan. Here's a straightforward version of the proof that relies on the facts that every left identity is also a right and that associativity holds in G.  Assume x' is a left inverse for a group element x and assume x'' is a right inverse. Let h a 2 sided identity in 
G (note we did NOT assume it's unique!It in fact is,but we haven't proven that yet! Be careful!) Then:
    x' = x'h = x'(xx'') = (x'x) x'' = hx''= x''. 
So x'=x'' and every left inverse of an element x is also a right. Q.E.D.     
