Editing Coset (section)

== Properties ==
The disjointness of non-identical cosets is a result of the fact that if {{mvar|x}} belongs to {{math|''gH''}} then {{math|1=''gH'' = ''xH''}}. For if {{math|''x'' ∈ ''gH''}} then there must exist an {{math|''a'' ∈ ''H''}} such that {{math|1=''ga'' = ''x''}}. Thus {{math|1=''xH'' = (''ga'')''H'' = ''g''(''aH'')}}. Moreover, since {{math|''H''}} is a group, left multiplication by {{mvar|a}} is a bijection, and {{math|1=''aH'' = ''H''}}.

Thus every element of {{math|''G''}} belongs to exactly one left coset of the subgroup {{math|''H''}},<ref name=Rotman2006 /> and {{math|''H''}} is itself a left coset (and the one that contains the identity).<ref name=Dean />

Two elements being in the same left coset also provide a natural [[equivalence relation]]. Define two elements of {{mvar|G}}, {{mvar|x}} and {{mvar|y}}, to be equivalent with respect to the subgroup {{mvar|H}} if {{math|1=''xH'' = ''yH''}} (or equivalently if {{math|''x''<sup>−1</sup>''y''}} belongs to {{mvar|H}}). The [[equivalence classes]] of this relation are the left cosets of {{mvar|H}}.<ref>{{harvnb|Rotman|2006|loc=p.155}}</ref> As with any set of equivalence classes, they form a [[Partition (set theory)|partition]] of the underlying set. A '''coset representative''' is a [[representative (mathematics)|representative]] in the equivalence class sense. A set of representatives of all the cosets is called a [[Transversal (combinatorics)|transversal]]. There are other types of equivalence relations in a group, such as conjugacy, that form different classes which do not have the properties discussed here.

Similar statements apply to right cosets.

If {{math|''G''}} is an [[abelian group]], then {{math|1=''g'' + ''H'' = ''H'' + ''g''}} for every subgroup {{math|''H''}} of {{math|''G''}} and every element {{mvar|g}} of {{math|''G''}}. For general groups, given an element {{mvar|g}} and a subgroup {{math|''H''}} of a group {{math|''G''}}, the right coset of {{math|''H''}} with respect to {{mvar|g}} is also the left coset of the [[conjugate subgroup]] {{math|''g''<sup>−1</sup>''Hg''}} with respect to {{mvar|g}}, that is, {{math|1=''Hg'' = ''g''(''g''<sup>−1</sup>''Hg'')}}.

=== Normal subgroups ===
A subgroup {{math|''N''}} of a group {{math|''G''}} is a [[normal subgroup]] of {{math|''G''}} if and only if for all elements {{mvar|g}} of {{math|''G''}} the corresponding left and right cosets are equal, that is, {{math|1=''gN'' = ''Ng''}}. This is the case for the subgroup {{mvar|H}} in the first example above. Furthermore, the cosets of {{math|''N''}} in {{math|''G''}} form a group called the [[quotient group|quotient group or factor group]] {{math|''G''{{hsp}}/{{hsp}}''N''}}.

If {{math|''H''}} is not [[normal subgroup|normal]] in {{math|''G''}}, then its left cosets are different from its right cosets. That is, there is an {{mvar|a}} in {{math|''G''}} such that no element {{mvar|b}} satisfies {{math|1=''aH'' = ''Hb''}}. This means that the partition of {{math|''G''}} into the left cosets of {{math|''H''}} is a different partition than the partition of {{math|''G''}} into right cosets of {{math|''H''}}. This is illustrated by the subgroup {{mvar|T}} in the first example above. (''Some'' cosets may coincide. For example, if {{mvar|a}} is in the [[center (group theory)|center]] of {{math|''G''}}, then {{math|1=''aH'' = ''Ha''}}.)

On the other hand, if the subgroup {{math|''N''}} is normal the set of all cosets forms a group called the quotient group {{math|''G''{{hsp}}/{{hsp}}''N''}} with the operation {{math|∗}} defined by {{math|1=(''aN'') ∗ (''bN'') = ''abN''}}. Since every right coset is a left coset, there is no need to distinguish "left cosets" from "right cosets".

=== Index of a subgroup ===
{{Main|Index of a subgroup}}
Every left or right coset of {{math|''H''}} has the same number of elements (or [[cardinality]] in the case of an [[Infinity|infinite]] {{math|''H''}}) as {{math|''H''}} itself. Furthermore, the number of left cosets is equal to the number of right cosets and is known as the '''index''' of {{math|''H''}} in ''G'', written as {{math|[''G'' : ''H'']}}. [[Lagrange's theorem (group theory)|Lagrange's theorem]] allows us to compute the index in the case where {{math|''G''}} and {{math|''H''}} are finite:
<math display="block">|G| = [G : H]|H|.</math>
This equation can be generalized to the case where the groups are infinite.