x* x=e, x is its own inverse
y *y= e, y is its own inverse
(x *y)*( x* y)= e, x *y is its own inverse
(y* x)* (y* x)= e, y* x is its own inverse
also x* x* e= e*e can be rewritten as follows
x* y* y *x= e *y* y* e= e, (Since y *y= e)
(x* y)* (y* x)= e shows that (x *y) and (y *x)
are each other’s inverse and we already know that
(x *y) and (y* x) are inverse of its own.
As per (G,*) to be group any element should have
only one inverse element (unique)
This implies x *y= y* x (is one element)
So the elements of such group are 4 which are {x, y,e,x *y }.