Let be groups for and let . Prove that the map given by for all is an isomorphism (so that changing the order of the factors in a finite direct product does not change the isomorphism type).
We need to show that is a bijective homomorphism.
- Homomorphism: Let and . Then for each ; hence .
- Injective: Let such that . Then for each , we have . Because is a permutation of , we have for each ; hence . Thus is injective.
- Surjective: Let . It is clear that, with , we have . Thus is surjective.