Editing Coset (section)

{{short description|Disjoint, equal-size subsets of a group's underlying set}}
{{distinguish|Cosette}}

[[File:Left cosets of Z 2 in Z 8.svg|thumb|{{mvar|G}} is the group <math>\mathbb{Z}/8\mathbb{Z}</math>, the [[Integers modulo n|integers mod 8]] under addition. The subgroup {{mvar|H}} contains only 0 and 4. There are four left cosets of {{mvar|H}}: {{mvar|H}} itself, {{math|1 + ''H''}}, {{math|2 + ''H''}}, and {{math|3 + ''H''}} (written using additive notation since this is the [[additive group]]). Together they partition the entire group {{mvar|G}} into equal-size, non-overlapping sets. The [[Index of a subgroup|index]] {{math|[''G'' : ''H'']}} is 4.]]

In [[mathematics]], specifically [[group theory]], a [[subgroup]] {{mvar|H}} of a [[group (mathematics)|group]] {{mvar|G}} may be used to decompose the underlying [[Set (mathematics)|set]] of {{mvar|G}} into [[disjoint sets|disjoint]], equal-size [[subset]]s called '''cosets'''. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) have the same number of elements ([[cardinality]]) as does {{mvar|H}}. Furthermore, {{mvar|H}} itself is both a left coset and a right coset. The number of left cosets of {{mvar|H}} in {{mvar|G}} is equal to the number of right cosets of {{mvar|H}} in {{mvar|G}}. This common value is called the [[Index of a subgroup|index]] of {{mvar|H}} in {{mvar|G}} and is usually denoted by {{math|[''G'' : ''H'']}}.

Cosets are a basic tool in the study of groups; for example, they play a central role in [[Lagrange's theorem (group theory)|Lagrange's theorem]] that states that for any [[finite group]] {{mvar|G}}, the number of elements of every subgroup {{mvar|H}} of {{mvar|G}} divides the number of elements of {{mvar|G}}. Cosets of a particular type of subgroup (a [[normal subgroup]]) can be used as the elements of another group called a [[quotient group|quotient group or factor group]]. Cosets also appear in other areas of mathematics such as [[vector space]]s and [[error-correcting code]]s.