Normal Subgroups and Quotient Groups

A subgroup $N \leq G$ is normal (written $N \trianglelefteq G$) if $gNg^{-1} = N$ for all $g \in G$ I.e., $gng^{-1} \in N$ for all $g \in G$ and all $n \in N$.

Proposition 4.1. Every subgroup of an abelian group is normal.

Proposition 4.2. The following are equivalent for $N \leq G$:

1. $N \trianglelefteq G$.
2. $gN = Ng$ for all $g \in G$ (left and right cosets coincide).
3. The product of two left cosets is again a left coset: $(aN)(bN) = (ab)N$.

Proof of (1) $\Rightarrow$ (3). Let $a_1n_1 \in aN$ and $b_1 n_2 \in bN$. Then $(a_1 n_1)(b_1 n_2) = a_1 b_1 (b_1^{-1} n_1 b_1) n_2$. Since $N$ is normal, $b_1^{-1} n_1 b_1 \in N$ So $(b_1^{-1} n_1 b_1) n_2 \in N$Giving $(a_1 n_1)(b_1 n_2) \in a_1 b_1 N$. $\blacksquare$

When $N \trianglelefteq G$The set $G/N = \{gN : g \in G\}$ of cosets forms a group under

$(aN)(bN) = (ab)N$

Called the quotient group of $G$ by $N$.

Theorem 4.3. If $G$ is finite and $N \trianglelefteq G$ Then $|G/N| = [G : N] = |G|/|N|$.

Example. $S_3 / A_3 \cong \mathbb{Z}/2\mathbb{Z}$.

Example. $\mathbb{Z}/n\mathbb{Z}$ is the quotient of $\mathbb{Z}$ by $n\mathbb{Z}$.

Problem. The quaternion group $Q_8 = \{1, -1, i, -i, j, -j, k, -k\}$ has center $Z(Q_8) = \{1, -1\}$. Compute $Q_8 / Z(Q_8)$.

Problem. Let $N = \langle (1\ 2)(3\ 4), (1\ 3)(2\ 4) \rangle \leq S_4$. Show that $N \trianglelefteq S_4$ and identify $S_4/N$.

Problem. Define $\phi : GL_2(\mathbb{R}) \to \mathbb{R}^*$ by $\phi(A) = \det(A)$. Identify $\ker(\phi)$ and $GL_2(\mathbb{R})/\ker(\phi)$.

Problem. Show that $\mathbb{C}^* / S^1 \cong \mathbb{R}^+$Where $S^1 = \{z \in \mathbb{C}^* : |z| = 1\}$.

A normal subgroup is a subgroup that is invariant under conjugation: $gNg^{-1} = N$ for all $g \in G$. This means the subgroup “looks the same” from every perspective in the group. Normality is the algebraic condition that makes quotient groups possible: when $N$ is normal, the cosets $G/N$ inherit a group structure because the product of two cosets is well-defined.

The quotient group $G/N$ collapses all elements of $N$ to the identity, creating a simpler group that captures the “large-scale” structure of $G$ while ignoring the internal structure of $N$. The first isomorphism theorem says that $G/\ker(\phi) \cong \mathrm{im}(\phi)$: every homomorphism factors through its quotient. This means quotient groups are the natural objects that arise from homomorphisms. For example, $\mathbb{Z}/n\mathbb{Z}$ is the quotient that collapses all multiples of $n$ to zero, creating a finite cyclic group. The cosets of a normal subgroup partition the group into equal-sized pieces, and the quotient group describes how these pieces fit together.

• Forgetting to check all cosets. When verifying normality via $gNg^{-1} \subseteq N$, you must check every $g \in G$, not just generators.
• Confusing $G/N$ with $N/G$. The quotient $G/N$ is defined only when $N \trianglelefteq G$; the notation is not symmetric.
• Assuming all subgroups are normal. In non-abelian groups, most subgroups are not normal. For instance, $\langle (1\ 2) \rangle$ is not normal in $S_3$.
• Miscalculating coset products. Always reduce representatives: $(aN)(bN) = (ab)N$, but $(ab)N$ may simplify further if $ab$ can be rewritten.

flowchart TD
    A[4_Normal Subgroups And Quotient Groups] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

• Lagrange’s Theorem: Lagrange’s theorem constrains the index of a normal subgroup and the order of the quotient group.

• Homomorphisms and Isomorphism Theorems: The first isomorphism theorem shows that every quotient group arises from a homomorphism.

• Groups: The group axioms underpin the construction of normal subgroups and quotient groups.

• Quantum Mechanics

• Graph Theory

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Ensure you have mastered the prerequisite material before attempting this advanced content.