Prove all parts of the Lattice Isomorphism Theorem.
Let be a group and a normal subgroup, with the natural projection . Let denote the set of all subgroups of containing and the set of all subgroups of , and define by . Let . Then we have the following.
- is a bijection. (Injective) Suppose ; then . Now let . We have for some , so that . Then , and we have . Similarly, if then . Thus , and is injective. (Surjective) Let ; we saw previously that the preimage of a subgroup under a group homomorphism is a subgroup, so that there exists some such that .
- if and only if . If , then , so . Thus . Suppose and let . Now for some , so that . Then , hence .
- If , then . If , then by the previous part, we have partitions and and every equivalence class has the form or , respectively. Define by . (Well defined) If , then , so , hence . (Injective) Suppose . Then , and we have for some . Now , so that . Thus . (Surjective) is surjective because is surjective. Thus is a bijection and the conclusion follows.
- . Let . Then for some , so that for some . Then , where . Thus . If , then for some . Now , and for each . Thus for some , so that .
- . If , then for some . Now , so , and similarly , so . If , then for some and . Now , so that , hence . Thus .
- is normal if and only if is normal. Suppose is normal, and let . Because is surjective, we have for some . Now , so that is normal. Now suppose is normal and let . We have . Thus if , we have for some , so . Thus is normal.