8.2 Normal Subgroups and Quotient Groups

Normal subgroups and quotient groups are key concepts in abstract algebra. They allow us to create new groups from existing ones, simplifying complex structures. This topic builds on earlier ideas about subgroups and cosets, showing how they connect to form larger algebraic systems.

Understanding normal subgroups and quotient groups is crucial for grasping advanced group theory. These concepts help us break down groups into smaller parts, revealing hidden patterns and relationships. They're essential tools for analyzing group structures and solving complex algebraic problems.

Normal subgroups and quotient groups

Definition and properties of normal subgroups

• A subgroup $H$ of a group $G$ is normal if $gH = Hg$ for all $g \in G$
  • This means the left and right cosets of $H$ in $G$ are equal
  • Notation: $H \trianglelefteq G$ denotes that $H$ is a normal subgroup of $G$
• Equivalent condition: A subgroup $N$ of $G$ is normal if and only if $gNg^{-1} \subseteq N$ for all $g \in G$
  • This means conjugation by any element of $G$ maps $N$ to itself
  • Example: In the group of integers under addition $(\mathbb{Z}, +)$, any subgroup $n\mathbb{Z}$ (multiples of $n$) is normal

Definition and properties of quotient groups

• The quotient group (or factor group) of a group $G$ by a normal subgroup $N$, denoted $G/N$, is the set of all cosets of $N$ in $G$ with the group operation defined by $(aN)(bN) = (ab)N$ for $a, b \in G$
  • The elements of $G/N$ are the cosets of $N$ in $G$, and the group operation is well-defined because $N$ is normal in $G$
  • Example: The quotient group $\mathbb{Z}/2\mathbb{Z}$ has elements $\{0 + 2\mathbb{Z}, 1 + 2\mathbb{Z}\}$ (even and odd integers)
• The order of $G/N$ is equal to the index of $N$ in $G$, denoted $|G:N|$, which is the number of distinct cosets of $N$ in $G$
  • Formula: $|G/N| = |G:N| = |G| / |N|$
• The natural projection map $\pi: G \to G/N$ defined by $\pi(g) = gN$ is a surjective group homomorphism with kernel $N$

Properties of normal subgroups and quotient groups

Proofs involving normal subgroups

• Prove that the kernel of a group homomorphism is a normal subgroup of the domain group
  • Let $f: G \to H$ be a group homomorphism with kernel $K = \{g \in G \mid f(g) = e_H\}$
  • For any $g \in G$ and $k \in K$, $f(gkg^{-1}) = f(g)f(k)f(g^{-1}) = f(g)e_Hf(g^{-1}) = e_H$, so $gkg^{-1} \in K$
  • Thus, $K$ is normal in $G$
• Prove that the cosets of a normal subgroup $N$ in $G$ form a partition of $G$
  • The cosets are disjoint: If $aN \cap bN \neq \emptyset$, then $aN = bN$
  • The union of all cosets is equal to $G$: $\bigcup_{a \in G} aN = G$

Proofs involving quotient groups

• Prove that the quotient group $G/N$ is a well-defined group under the operation $(aN)(bN) = (ab)N$ for $a, b \in G$
  • Closure: For any $aN, bN \in G/N$, $(aN)(bN) = (ab)N \in G/N$
  • Associativity: Follows from the associativity of $G$
  • Identity: The coset $N = eN$ is the identity element in $G/N$
  • Inverses: For any $aN \in G/N$, $(aN)(a^{-1}N) = (aa^{-1})N = eN = N$
• Prove that the natural projection map $\pi: G \to G/N$ is a surjective group homomorphism with kernel $N$
  • Homomorphism: For any $a, b \in G$, $\pi(ab) = (ab)N = (aN)(bN) = \pi(a)\pi(b)$
  • Surjectivity: For any $aN \in G/N$, there exists $a \in G$ such that $\pi(a) = aN$
  • Kernel: $\ker(\pi) = \{g \in G \mid \pi(g) = N\} = \{g \in G \mid gN = N\} = N$

Constructing quotient groups

Constructing quotient groups from a group and normal subgroup

• Given a group $G$ and a normal subgroup $N$, construct the quotient group $G/N$ by:
  1. Determining the distinct cosets of $N$ in $G$
  2. Defining the group operation on these cosets using $(aN)(bN) = (ab)N$ for $a, b \in G$
• Example: Consider the group $\mathbb{Z}_4 = \{0, 1, 2, 3\}$ under addition modulo 4 and its subgroup $N = \{0, 2\}$
  • The cosets of $N$ in $\mathbb{Z}_4$ are $\{0, 2\}$ and $\{1, 3\}$
  • The quotient group $\mathbb{Z}_4/N$ has elements $\{N, 1+N\}$ with the operation $(a+N)+(b+N) = (a+b)+N$

Analyzing properties of quotient groups

• Determine the order of the quotient group $G/N$ using the formula $|G/N| = |G:N| = |G| / |N|$
  • Example: In the previous example, $|\mathbb{Z}_4/N| = |\mathbb{Z}_4| / |N| = 4 / 2 = 2$
• Construct the multiplication table for the quotient group $G/N$ using the group operation $(aN)(bN) = (ab)N$ for $a, b \in G$
• Identify the identity element (the coset $N$) and inverse elements in the quotient group $G/N$
  • Example: In $\mathbb{Z}_4/N$, the identity is $N$, and each element is its own inverse
• Determine whether the quotient group $G/N$ is abelian, cyclic, or has other special properties based on the properties of $G$ and $N$
  • Example: $\mathbb{Z}_4/N$ is abelian and cyclic, as it is isomorphic to $\mathbb{Z}_2$

Fundamental theorem of homomorphisms for quotient groups

Statement and application of the fundamental theorem of homomorphisms

• The fundamental theorem of homomorphisms states: If $f: G \to H$ is a group homomorphism with kernel $K$, then the image of $f$, denoted $\text{im}(f)$, is isomorphic to the quotient group $G/K$
  • In other words, $\text{im}(f) \cong G/\ker(f)$
  • This theorem establishes a relationship between quotient groups and homomorphic images
• Use the fundamental theorem to establish isomorphisms between quotient groups and subgroups of the codomain group
  • Example: If $f: \mathbb{Z} \to \mathbb{Z}_6$ is defined by $f(n) = [n]_6$ (remainder of $n$ divided by 6), then $\ker(f) = 6\mathbb{Z}$ and $\text{im}(f) \cong \mathbb{Z}/6\mathbb{Z}$

Solving problems using the fundamental theorem of homomorphisms

• Apply the fundamental theorem to determine the structure of quotient groups based on the properties of the domain group and the kernel of the homomorphism
  • Example: If $f: G \to H$ is a surjective homomorphism and $G$ is abelian, then $H \cong G/\ker(f)$ is also abelian
• Solve problems involving the order of elements, subgroups, and quotient groups using the fundamental theorem and the properties of isomorphisms
  • Example: If $f: G \to H$ is a surjective homomorphism and $|G| = 60$, $|\ker(f)| = 10$, then $|H| = |G|/|\ker(f)| = 60/10 = 6$
• Use the fundamental theorem to prove that certain groups are isomorphic to quotient groups of other groups
  • Example: The cyclic group $\mathbb{Z}_n$ is isomorphic to the quotient group $\mathbb{Z}/n\mathbb{Z}$, as the natural projection $\pi: \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ is a surjective homomorphism with kernel $n\mathbb{Z}$