Editing Quotient group (section)

{{Short description|Group obtained by aggregating similar elements of a larger group}}
{{Group theory sidebar |Basics}}

A '''quotient group''' or '''factor group''' is a [[math]]ematical [[group (mathematics)|group]] obtained by aggregating similar elements of a larger group using an [[equivalence relation]] that preserves some of the group structure (the rest of the structure is "factored out").  For example, the [[cyclic group]] of [[modular arithmetic|addition modulo ''n'']] can be obtained from the group of [[integer]]s under addition by identifying elements that differ by a multiple of <math>n</math> and defining a group structure that operates on each such class (known as a [[congruence class]]) as a single entity. It is part of the mathematical field known as [[group theory]].

For a [[congruence relation]] on a group, the [[equivalence class]] of the [[identity element]] is always a [[normal subgroup]] of the original group, and the other equivalence classes are precisely the [[coset]]s of that normal subgroup.  The resulting quotient is written {{tmath|1= G\,/\,N }}, where <math>G</math> is the original group and <math>N</math> is the normal subgroup.  This is read as '{{tmath|1= G\bmod N }}', where <math>\text{mod}</math> is short for [[modular arithmetic|modulo]].  (The notation {{tmath|1= G\,/\,H }} should be interpreted with caution, as some authors (e.g., Vinberg<ref>{{cite book |last=Vinberg |first=Ė B. |title=A course in algebra |date=2003 |publisher=American Mathematical Society |isbn=978-0-8218-3318-6 |series=Graduate studies in mathematics |location=Providence, R.I |pages=157}}</ref>) use it to represent the left cosets of <math>H</math> in <math>G</math> for ''any'' subgroup <math>H</math>, even though these cosets do not form a group if <math>H</math> is not normal in {{tmath|1= G }}.  Others (e.g., Dummit and Foote<ref>{{harvtxt|Dummit|Foote|2003|p=95}}</ref>) use this notation to refer only to the quotient group, with the appearance of this notation implying that <math>H</math> is normal in {{tmath|1= G }}.)

Much of the importance of quotient groups is derived from their relation to [[group homomorphism|homomorphisms]].  The [[Isomorphism_theorems#First_isomorphism_theorem|first isomorphism theorem]] states that the [[image (mathematics)|image]] of any group <math>G</math> under a homomorphism is always [[group isomorphism|isomorphic]] to a quotient of {{tmath|1= G }}.  Specifically, the image of <math>G</math> under a homomorphism <math>\varphi: G \rightarrow H</math> is isomorphic to <math>G\,/\,\ker(\varphi)</math> where <math>\ker(\varphi)</math> denotes the [[kernel (algebra)#Group homomorphisms|kernel]] of {{tmath|1= \varphi }}.

The [[duality (mathematics)|dual]] notion of a quotient group is a [[subgroup]], these being the two primary ways of forming a smaller group from a larger one. Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup. In [[category theory]], quotient groups are examples of [[quotient object]]s, which are [[dual (category theory)|dual]] to [[subobject]]s.

{{for|other examples of quotient objects|quotient ring|quotient space (linear algebra)|quotient space (topology)|quotient set}}