rightcoset

In group theory, a right coset of a subgroup H of a group G by an element g ∈ G is the subset Hg = { h g : h ∈ H } of G. The corresponding left coset is gH. For g varying over G, the collection { Hg : g ∈ G } partitions G into equal-sized subsets, each of cardinality |H|. Distinct right cosets are disjoint, and the set of all right cosets is denoted G/H, called the right coset space or quotient set.

Two elements g1, g2 ∈ G lie in the same right coset if and only if g1 ∈ H g2, equivalently g1 g2^{-1} ∈ H. Thus the cosets form an equivalence relation on G with respect to H.

If H is a normal subgroup of G (written H ⫽ G), then Hg = gH for all g ∈ G, and the quotient set G/H inherits a group structure. The product of cosets is defined by (Hg1)(Hg2) = H(g1 g2), making G/H a quotient group. In this case, left and right cosets coincide.

The index [G:H] is the number of right cosets. If G is finite, |G/H| = [G:H] = |G|/|H|.

Example: In the additive group of integers Z, let H = nZ for some n > 0. The right coset a + nZ equals the residue class of a modulo n; there are exactly n such cosets, and they partition Z.

Right cosets provide a natural partition of G, a way to form quotient structures when normality holds, and a foundation for many results in group theory, including Lagrange’s theorem and actions on coset spaces.