Quotient group: Difference between revisions

Latest revision as of 15:28, 9 April 2026

A quotient group is a group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure.

Definitions

Let $G$ be a group and $N ⊴ G$ a normal subgroup.

Definition via cosets

The quotient group $G / N$ is the set of left cosets $G / N : = {g N ∣ g \in G}$ with group operation $(g N) (h N) : = (g h) N .$

Definition via equivalence relations

Define a relation on $G$ by $g \sim h ⟺ g^{- 1} h \in N .$

The relation $\sim$ is a equivalence relation, and denote $[g] : = {a ∣ a \sim g}$ as the equivalence class of $g$ . The quotient group is defined by the set of equivalence classes $G / N : = G / \sim$ with operation $[g] [h] : = [g h] .$

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-A '''quotient group''' is a group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure.
+A '''quotient group''' is a [[group]] obtained by aggregating similar elements of a larger group using an [[equivalence relation]] that preserves some of the group structure.
 == Definitions ==
-Let <math>G</math> be a group and <math>N\trianglelefteq G</math> a normal subgroup.
+Let <math>G</math> be a group and <math>N\trianglelefteq G</math> a [[normal subgroup]].
 === Definition via cosets ===
-The quotient group <math>G/N</math> is the set of left cosets <math display="block">G/N:=\{gN\mid g\in G\}</math>with group operation <math>(gN)(hN):=(gh)N.</math>
+The quotient group <math>G/N</math> is the set of left [[Coset|cosets]] <math display="block">G/N:=\{gN\mid g\in G\}</math>with group operation <math>(gN)(hN):=(gh)N.</math>
 === Definition via equivalence relations ===
-Define a relation on <math>G</math> by<math display="block">g\sim h\Longleftrightarrow g^{-1}h\in N.</math>
+Define a [[relation]] on <math>G</math> by<math display="block">g\sim h\Longleftrightarrow g^{-1}h\in N.</math>
-The relation <math>\sim</math> is a equivalence relation, and denote <math>[g]:=\{a\mid a\sim g\}</math> as the equivalence class of <math>g</math>. The quotient group is defined by the set of equivalence classes<math display="block">G/N:=G/\sim</math>with operation <math>[g][h]:=[gh].</math>
+The relation <math>\sim</math> is a equivalence relation, and denote <math>[g]:=\{a\mid a\sim g\}</math> as the equivalence class of <math>g</math>. The quotient group is defined by the set of [[Equivalence relation#Equivalence classes|equivalence classes]]<math display="block">G/N:=G/\sim</math>with operation <math>[g][h]:=[gh].</math>