InfernalAtom683 at 07:28, 9 April 2026

2026-04-09T07:28:12Z

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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 ==		== 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 ===		=== 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 ===		=== 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>

InfernalAtom683: Created page with "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 $N\trianglelefteq G$ a normal subgroup. === Definition via cosets === The quotient group $G/N$ is the set of left cosets $G/N:=\{gN\mid g\in G\}$ with group operation $(gN)(hN):=(gh)N.$ === Defi..."

2026-04-05T08:18:51Z

Created page with "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. === 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> === Defi..."

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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.

== Definitions ==
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>

=== 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>

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>

Quotient group - Revision history

InfernalAtom683 at 07:28, 9 April 2026