Homomorphisms and Quotient Groups - Examples and Constructions

The isomorphism theorems provide powerful tools for understanding the structure of quotient groups and their relationship to subgroups and homomorphic images.

TheoremFirst Isomorphism Theorem

Let $\phi: G \to H$ be a group homomorphism. Then: $G/\ker(\phi) \cong \text{im}(\phi)$

The isomorphism is given by $\overline{\phi}: G/\ker(\phi) \to \text{im}(\phi)$ where $\overline{\phi}(g\ker(\phi)) = \phi(g)$ .

This fundamental result states that every homomorphism factors through a quotient by its kernel. The quotient "removes the redundancy" captured by the kernel, yielding an isomorphism to the image.

ExampleApplication of First Isomorphism Theorem

Consider the sign homomorphism $\text{sgn}: S_n \to \{\pm 1\}$ . We have:

$\ker(\text{sgn}) = A_n$ (the alternating group)
$\text{im}(\text{sgn}) = \{\pm 1\} \cong \mathbb{Z}_2$

Therefore: $S_n/A_n \cong \mathbb{Z}_2$

This shows that $S_n$ has exactly two cosets modulo $A_n$ : the even permutations $A_n$ and the odd permutations.

TheoremSecond Isomorphism Theorem (Diamond Theorem)

Let $H \leq G$ and $N \trianglelefteq G$ . Then: $H/(H \cap N) \cong (HN)/N$

where $HN = \{hn : h \in H, n \in N\}$ is the product of subgroups.

TheoremThird Isomorphism Theorem (Correspondence Theorem)

Let $N \trianglelefteq G$ . There is a bijection between:

Subgroups of $G$ containing $N$
Subgroups of $G/N$

given by $H \mapsto H/N$ . Under this correspondence, normal subgroups correspond to normal subgroups, and: $(G/N)/(H/N) \cong G/H$

Remark

The Third Isomorphism Theorem establishes that the lattice structure of subgroups is preserved under quotients. Subgroups "above" $N$ in $G$ correspond exactly to subgroups in $G/N$ . This is crucial for understanding the subgroup structure of quotient groups.

These theorems provide a systematic framework for working with quotients. Together they show that quotient constructions behave predictably with respect to subgroups and homomorphisms. The First Isomorphism Theorem in particular is used constantly to establish isomorphisms by constructing appropriate homomorphisms and computing their kernels.