Isomorphism Theorems

First Isomorphism Theorem

Proposition

Let be a group homomorphism, and the canonical quotient map. There is a well-defined map given by such that , in other words the following diagram commutes:

First Isomorphism Theorem

Let be a group homomorphism. Then . Specifically, the map given by is a group isomorphism.

Proof Let . We saw is well-defined. It is a homomorphism because The map is clearly surjective as is surjective. Now compute its kernel:

Second Isomorphism Theorem

Second Isomorphism Theorem

Let be a group, a subgroup and a normal subgroup.Then .

Proof

Third Isomorphism Theorem

Thrm Third Isomorphism Theorem Let be a group and normal subgroups with . Then . Proof

The Correspondence Theorem

Thrm Correspondence Theorem Let be a surjective group homomorphism with kernel . Then for every subgroup the inverse image is a subgroup of containing . The map sets up a bijection between the setsUnder this correspondence , and if and only if

Proof See [1, Theorem 2.10.5] or [2, Theorem 2.7.2] or [3, Prop 2.76].

Remark

The correspondence theorem says, in particular, that subgroups in a quotient are always of the form for a subgroup .

Prop Let be a group and let be two normal subgroups with and . Then . Proof