Coset in the context of Error-correcting code

⭐ Core Definition: Coset

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 codes.

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⭐ Core Definition: Coset
Coset in the context of Group homomorphism
Coset in the context of Index of a subgroup

Coset in the context of Group homomorphism

In mathematics, given two groups, (G,∗) and (H, ·), a group homomorphism from (G,∗) to (H, ·) is a function h : G → H such that for all u and v in G it holds that

where the group operation on the left side of the equation is that of G and on the right side that of H.

View the full Wikipedia page for Group homomorphism

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Coset in the context of Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G.The index is denoted $|G:H|$ or $[G:H]$ or $(G:H)$ .Because G is the disjoint union of the left cosets and because each left coset has the same size as H, the index is related to the orders of the two groups by the formula