Subgroup lattice and quotient lattice of finite abelian group are isomorphic
Statement in terms of anti-automorphism of the subgroup lattice
There is a set map such that:
- is isomorphic to .
- is the identity map.
The map is not canonical, but it is canonical up to pre-composition by automorphisms.
Note that (2) in particular shows that is the kernel of an endomorphism of with image the subgroup (specifically, take the quotient map to and compose with the isomorphism to . Thus, it shows that every subgroup of a finite abelian group is an endomorphism kernel. Similarly, it shows that every subgroup of a finite abelian group is an endomorphism image.
Denote by the Pontryagin dual of , i.e., the group of homomorphisms from to the circle group under pointwise multiplication.
By fact (1), is isomorphic to . Let be one such isomorphism.
Then, we define:
We now check the three conditions:
- : If , then any element of that has in its kernel also has in its kernel. Thus, . Applying preserves the containment.
- is isomorphic to : is the group of homomorphisms from to the circle group with in the kernel. These are naturally identified with the group of homomorphisms from to the circle group, which is . By fact (1), this is isomorphic to . Thus, is isomorphic to .
- is an automorphism: First, consider the map . Define the map as the analogue of from to . Denote by the map induced by on . Then, . Then, , which is the identity map.