Let G be a group with identity elements e and f. We want to prove that e = f.
By definition of identity element, for any element a in G, we have:
a · e = a and f · a = a
Let’s use e as the identity element in the first equation, and f as the identity element in the second equation:
a · e = a a · f = a
Now, we can use the associativity property of groups to rewrite the second equation:
a · f = (a · e) · f
Since e is the identity element, we can simplify the right-hand side to just a · f = a:
a · f = a
Comparing this equation with the one above it, we see that a · e = a · f. This means that:
a · e · f⁻¹ = a · f · f⁻¹
Using the associative property again, we can simplify this to:
a · e = a · f
But we already know that this equation holds for all elements a in G. Therefore, we must have e = f, and the identity element in a group is unique.