Let be a group and let be the set of all isomorphisms . Prove that is a group under function composition.
We need to verify that the three group axioms are satisfied: associativity, identity, and inverses.
- We know from set theory that function composition is always associative.
- Note that is a bijection and trivially a homomorphism, so that . Finally, we have for all isomorphisms , so that is an identity element under composition.
- Given , we know from set theory that an inverse exists. This inverse is a homomorphism, as we show. If , then . Since is injective, we have . Thus is a homomorphism, and we have .
Thus is a group under function composition.