Homomorphisms and Quotient Groups - Key Properties

Normal subgroups are distinguished subgroups that allow the construction of quotient groups, providing a systematic way to build new groups from old ones.

DefinitionNormal Subgroup

A subgroup $N$ of $G$ is normal (written $N \trianglelefteq G$ ) if it is invariant under conjugation: $gNg^{-1} = N \quad \text{for all } g \in G$

Equivalently, $gN = Ng$ for all $g \in G$ (left and right cosets coincide).

Every subgroup of an abelian group is automatically normal. For non-abelian groups, normality is a special property. The kernel of any homomorphism is always normal, and conversely, every normal subgroup arises as the kernel of some homomorphism.

DefinitionQuotient Group

Let $N \trianglelefteq G$ be a normal subgroup. The quotient group $G/N$ (read "G mod N") is the set of cosets $\{gN : g \in G\}$ with operation: $(aN)(bN) = (ab)N$

This operation is well-defined precisely because $N$ is normal. The quotient group has order $|G/N| = [G:N] = |G|/|N|$ .

ExampleQuotient Groups

Integers modulo n: $\mathbb{Z}/n\mathbb{Z} = \mathbb{Z}_n$ (cosets of $n\mathbb{Z}$ in $\mathbb{Z}$ )
Alternating group: $S_n/A_n \cong \mathbb{Z}_2$ (quotient by even permutations)
Circle group: $\mathbb{R}/\mathbb{Z} \cong S^1$ (the unit circle in $\mathbb{C}$ )
Projective space: $GL_n(\mathbb{R})/Z \cong PGL_n(\mathbb{R})$ where $Z$ is the center

The quotient construction "collapses" $N$ to the identity, treating all elements of $N$ as equivalent. The canonical projection $\pi: G \to G/N$ given by $\pi(g) = gN$ is always a surjective homomorphism with $\ker(\pi) = N$ .

Remark

Quotient groups formalize the idea of "modding out" by a subgroup. They appear throughout mathematics: modular arithmetic is the quotient $\mathbb{Z}/n\mathbb{Z}$ , and covering spaces in topology correspond to quotients of the fundamental group.

The relationship between normal subgroups and homomorphisms is fundamental. A subgroup $N$ is normal if and only if it is the kernel of some homomorphism. This correspondence is at the heart of the isomorphism theorems, which describe how quotients, subgroups, and homomorphic images interact.