Question

If (G,*) be a group ,then the inverse of each element of G is unique

Answer 1

Let (G,*) be a group and g∈G.
We have to prove that inverse of 'g' is unique.
Suppose that 'g' has inverse elements 'x' and 'y' . We want to show that x = y is the only possibility.
Let 'e' be the identity element in G.
Now, we know that,
g * x = e = x * g   .......... (1)
         and
g * y = e = y * g   .......... (2)
Consider the element   
         y * g * x
From the L.H.S of equation (1),
g * x = e
So, y * (g * x) = y * e = y   ........ (3)
(since 'e' is the identity)
From the R.H.S of equation (2),
y * g = e
(y * g) * x = e * x = x   ......... (4)
(since 'e' is the identity)
We know that,
y * (g * x) = (y * g) * x
From equation (3), L.H.S = y
From equation (4), R.H.S = x
Thus y = x
So, 'g' has a unique inverse in G.
Hence proved.