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Coset - Wikipedia

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In mathematics, specifically group theory, a subgroup H of a group G may be used to decompose the underlying set of G into disjoint, equal-size subsets called cosets. There are left cosets and right cosets. Cosets (both left and right) have the same number of elements (cardinality) as does H. Furthermore, H itself is both a left coset and a right coset. The number of left cosets of H in G is equal to the number of right cosets of H in G. This common value is called the index of H in G and is usually denoted by [G : H]. Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group G, the number of elements of every subgroup H of G divides the number of elements of G. Cosets of a particular type of subgroup (a normal subgroup) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as vector spaces and error-correcting cod

Coset - Wikipedia Jump to content From Wikipedia, the free encyclopedia Disjoint, equal-size subsets of a group's underlying set Not to be confused with Cosette . G is the group Z / 8 Z {\displaystyle \mathbb {Z} /8\mathbb {Z} } , the integers mod 8 under addition. The subgroup H contains only 0 and 4. There are four left cosets of H : H itself, 1 + H , 2 + H , and 3 + H (written using additive notation since this is the additive group ). Together they partition the entire group G into equal-size, non-overlapping sets. The index [ G  : H ] is 4. In mathematics , specifically group theory

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