Hello together!

I need to show whether the following is a(n abelian) group or not:

$\displaystyle G = \{w, f\}; a*b := (a \leftrightarrow b)$

I try to show the axioms.

1) Closure

Is given, since

w*f = f

f*w = f

w*w = w

f*f = f

2.) Associativity

???

3.) Identity

I need an element e with e*a = a*e = a. (for all elements of G)

So..

a*e = a <-> e = w.

because w*w = w and f*f = f.

4.) Inverse element;

I need an element i with

a*i = i*a = e.

the inverse for w is w, since w*w=w=e.

the inverse for f is f, since f*f=w=e.

5.) Commutativity:

Is a*b=b*a ?

This can be shown be the few examples:

w*f = f

f*w = f