Coset

A coset is a translate of a subgroup within a group, formed by multiplying every subgroup element by a fixed group element. Left and right cosets can differ when the group is nonabelian.

Left coset: A set formed by multiplying every element of a subgroup on the left by one fixed group element. Left multiplication defines one of the two standard ways to form a coset.
Group (mathematics): A set with an associative binary operation, an identity element, and inverses for every element. Cosets are formed inside groups using their operation.
Lagrange's theorem for groups: The theorem that the order of a subgroup of a finite group divides the group's order. Coset counting is the central argument behind this finite-group theorem.
Conjugate subgroup: A subgroup obtained by conjugating every element of another subgroup by a fixed group element. Conjugation changes the subgroup itself; coset translation moves its elements without changing it.
Right coset: A set formed by multiplying every element of a subgroup on the right by one fixed group element. Right multiplication may produce a different set when the group is nonabelian.
Subgroup: A subset of a group that is itself a group under the inherited operation. The subgroup supplies the elements being translated to form a coset.
Orbit-stabilizer theorem: A theorem equating an orbit's size with a group's order divided by its stabilizer's order, for finite groups. The orbit can be identified with cosets of the point stabilizer.
Coset representative: An element chosen from a coset to stand for that coset. Different representatives can describe the same coset, unlike distinct cosets.
Lagrange's theorem: A finite subgroup's order divides the order of its ambient group. Equal-sized cosets partition finite groups, yielding this divisibility result.
Group action: A way for group elements to act as transformations on a set while respecting group multiplication. Multiplication translates a subgroup as a set within the group.

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