Normal Subgroups, Quotient Groups, and Group Actions

April 11, 2026 · Luciano Muratore

1. The First Isomorphism Theorem

The First Isomorphism Theorem is a fundamental tool in group theory that relates a group, its image under a homomorphism, and its kernel.

• Let $\varphi:G\rightarrow H$ be a group homomorphism.
• Theorem: $G/ker(\varphi)\cong Im(\varphi)$.
• If the homomorphism is surjective (onto), then $G/ker(\varphi)\cong H$.

Example: $\mathbb{R}/\mathbb{Z}\cong U$

• Define the map $\varphi:\mathbb{R}\rightarrow U$ by $\varphi(x)=e^{2\pi ix}$.
• The kernel is $ker(\varphi)=\mathbb{Z}$.
• The image is $Im(\varphi)=U$.
• By the First Isomorphism Theorem, the quotient of the real numbers by the integers is isomorphic to the unit circle $U$.

2. Finite Order Elements

In the group $U$ (the unit circle), we can identify elements of finite order.

• An element $e^{2\pi ix} \in U$ has finite order if and only if $x \in \mathbb{Q}$.
• Let $V = \{e^{2\pi iq} \mid q \in \mathbb{Q}\}$ be the set of all such elements.
• This group of roots of unity is isomorphic to the quotient of the rationals by the integers: $V \cong \mathbb{Q}/\mathbb{Z}$.

3. Product Groups and Quotients

When dealing with product groups $G \times H$, we can define projections $\pi_1$ and $\pi_2$.

Projections and Kernels

• $\pi_1(g,h)=g$ with $ker(\pi_1)=\{e_G\}\times H$.
• $\pi_2(g,h)=h$ with $ker(\pi_2)=G\times\{e_H\}$.

Quotient Properties

• The quotient by a factor is isomorphic to the other factor: $(G\times H)/(\{e_G\}\times H)\cong G$.
• For subgroups $A \le G$ and $B \le H$, the quotient of the products is the product of the quotients: $(G\times H)/(A\times B) \cong (G/A) \times (H/B)$

4. Linear Groups: $GL_n(K)$ and $SL_n(K)$

The relationship between general and special linear groups is a classic application of quotient groups.

• $GL_n(K)$: The group of invertible $n \times n$ matrices.
• $SL_n(K)$: The group of matrices with determinant 1.
• The determinant map $det: GL_n(K) \rightarrow K$ is a homomorphism where the kernel is $SL_n(K)$.
• Therefore, $GL_n(K)/SL_n(K) \cong K^{\times}$.

5. Centers and Inner Automorphisms

The Center of a Group

The center $Z(G)$ consists of all elements that commute with every element in $G$: $Z(G) = \{z \in G \mid zg = gz, \forall g \in G\}$

Inner Automorphisms

An inner automorphism $\phi_g$ is defined by conjugation: $\phi_g(x) = gxg^{-1}$.

• The set of all inner automorphisms is denoted $Int(G)$.
• The map $C: G \rightarrow Int(G)$ is a homomorphism with $ker(C) = Z(G)$.
• Isomorphism: $G/Z(G) \cong Int(G)$.

6. Group Actions

A group action of $G$ on a set $X$ is a map $G \times X \rightarrow X$ satisfying $e \cdot x = x$ and $g \cdot (h \cdot x) = (gh) \cdot x$.

Key Concepts

• Orbit ($O_x$): The set of points $\{g \cdot x \mid g \in G\}$.
• Fixed Points ($X^g$): The set of points $\{x \in X \mid g \cdot x = x\}$.

Types of Actions

• Faithful: Only the identity element acts as the identity map.
• Free: No non-identity element fixes any point.
• Transitive: There is only one orbit (any point can be moved to any other point).
• Simply Transitive: An action that is both free and transitive.

Conclusion

• Quotients arise naturally from homomorphisms.
• Centers control the structure of inner automorphisms.
• Group Actions encode symmetry and their orbits classify the behavior of the action.