3.3 Normal Subgroups and Quotient Groups

Normal subgroups are special subgroups that play a crucial role in group theory. They allow us to create quotient groups, which help simplify complex group structures and reveal important relationships between different groups.

Understanding normal subgroups and quotient groups is key to grasping advanced concepts in group theory. These ideas are essential for studying group homomorphisms, analyzing group structures, and exploring connections between algebra and geometry in mathematics.

Normal subgroups and their significance

Definition and properties of normal subgroups

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• Normal subgroups H of a group G satisfy $gHg^{-1} = H$ for all $g \in G$
  • $gHg^{-1}$ represents the set of all elements $ghg^{-1}$ with $h \in H$
• Equivalent condition states $gH = Hg$ for all $g \in G$
  • Left and right cosets of H in G are identical
• Improper normal subgroups always include
  • Trivial subgroup $\{e\}$
  • Group G itself
• All subgroups in abelian groups are normal due to commutativity
• Examples of normal subgroups
  • Center of a group is always a normal subgroup
  • Alternating group $A_n$ is a normal subgroup of the symmetric group $S_n$ for $n \geq 3$

Importance in group theory

• Allow construction of quotient groups
• Fundamental to the study of group homomorphisms
• Key role in group classification and structure analysis
  • Particularly important for simple groups and composition series
• Enable factor group construction
  • Essential for studying group extensions
  • Used in classification of finite simple groups
• Crucial in understanding Sylow subgroups
  • Fundamental to proving Sylow's theorems
• Connect algebraic and geometric aspects of group theory
  • Normal subgroups are precisely the kernels of group homomorphisms

Proving normality of subgroups

Methods for proving normality

• Show $gHg^{-1} \subseteq H$ for all $g \in G$
  • Equivalent to demonstrating $gHg^{-1} = H$ for all $g \in G$
• Prove $gH = Hg$ for all $g \in G$
• For finite groups, show $|gH| = |Hg|$ for all $g \in G$
• Apply conjugation test
  • H is normal in G if and only if $ghg^{-1} \in H$ for all $g \in G$ and $h \in H$
• Prove kernel of a group homomorphism is normal
• Leverage specific group properties (abelian, cyclic) to simplify proofs

Examples of normality proofs

• Prove normality of subgroup H = {1, -1} in group G = {1, -1, i, -i} under multiplication
  • Show $gHg^{-1} = H$ for all $g \in G$
  • Example: $(i)(1)(-i) = i(-i) = 1 \in H$ and $(i)(-1)(-i) = -1 \in H$
• Demonstrate normality of subgroup 2Z in group Z under addition
  • Prove $a + 2Z = 2Z + a$ for all $a \in Z$
  • Example: $3 + 2Z = \{..., 1, 3, 5, ...\} = 2Z + 3$

Constructing quotient groups

Formation and properties of quotient groups

• Quotient group G/H formed by set of all cosets of H in G
  • Group operation defined as $(aH)(bH) = (ab)H$
• Identity element of G/H is coset H itself
• Inverse of aH is $a^{-1}H$
• Order of G/H equals index of H in G
  • For finite groups, $|G/H| = |G|/|H|$
• First Isomorphism Theorem states G/ker(φ) isomorphic to image of φ
  • For group homomorphism φ: G → G'
• Quotient groups preserve certain group properties
  • May preserve being abelian or cyclic
  • May not preserve being simple
• Correspondence Theorem establishes bijection
  • Between subgroups of G/H and subgroups of G containing H
  • Preserves normality and other properties

Examples of quotient group constructions

• Construct Z/4Z quotient group
  • Cosets: 0 + 4Z, 1 + 4Z, 2 + 4Z, 3 + 4Z
  • Addition table: $(1 + 4Z) + (2 + 4Z) = 3 + 4Z$
• Form quotient group of real numbers R by integers Z
  • R/Z represents angles on a circle
  • Addition modulo 1: $0.7 + 0.8 \equiv 0.5 \pmod{1}$

Applying normal subgroups and quotient groups to homomorphisms

Relationship between homomorphisms and normal subgroups

• Kernel of group homomorphism φ: G → G' always normal subgroup of G
• Image of φ isomorphic to G/ker(φ)
• Normal subgroups are precisely kernels of group homomorphisms
• First Isomorphism Theorem applications
  • Construct isomorphisms between groups
  • Simplify complex group structures
• Second and Third Isomorphism Theorems
  • Analyze structure of quotient groups
  • Explore relationships between quotient groups

Examples of applications in group theory

• Use First Isomorphism Theorem to prove Z/nZ isomorphic to cyclic subgroup of order n in any group
• Analyze structure of dihedral groups using quotient groups
  • Example: $D_8/\{1, r^2\}$ isomorphic to $V_4$ (Klein four-group)
• Apply quotient groups in Sylow theory
  • Example: If P is a Sylow p-subgroup of G, then NG(P)/P is isomorphic to a subgroup of Aut(P)