Prove that the identity element in a group is unique.

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.