rechtercosets

A right coset is a fundamental concept in group theory, a branch of abstract algebra. It is a subset of a group that is generated by multiplying a fixed subgroup by a fixed element from the group. Specifically, given a group G and a subgroup H of G, the right coset of H in G with respect to an element g in G is the set of all elements of G that can be written as the product of an element of H and g. This is denoted as Hg = {hg | h ∈ H}.

Right cosets are used to partition the group G into disjoint subsets, each of which has the same size as the subgroup H. This partition is known as the right coset decomposition of G with respect to H. The number of distinct right cosets of H in G is equal to the index of H in G, denoted as [G:H], which is the ratio of the order of G to the order of H.

Right cosets are closely related to the concept of equivalence relations. Specifically, the relation defined by g ~ k if and only if g⁻¹k ∈ H is an equivalence relation on G, and the equivalence classes of this relation are precisely the right cosets of H in G.

Right cosets are also used in the study of group actions and in the classification of groups. For example, the number of distinct right cosets of a normal subgroup H in a group G is equal to the order of the quotient group G/H. Additionally, the right cosets of a subgroup H in a group G can be used to construct a group action of G on the set of right cosets of H, which is known as the right regular representation of G.