coset of a group G that possesses a subgroup H. A coset of G modulo H determined by the element x of G is a subset: x ◦ H = {x ◦ h | h ∈ H} H ◦ x = {h ◦ x | h ∈ H}

where ◦ is the dyadic operation defined on G. A subset of the former kind is called a left coset of G modulo H or a left coset of G in H; the latter is a right coset. In special cases x ◦ H = H ◦ x

for any x in G. Then H is called a normal subgroup of G. Any subgroup of an abelian group is a normal subgroup.

The cosets of G in H form a partition of the group G, each coset showing the same number of elements as H itself. These can be viewed as the equivalence classes of a left coset relation defined on the elements g₁ and g₂ of G as follows: g₁ ρ g₂ iff g₁ ◦ H = g₂ ◦ H

Similarly a right coset relation can be defined. When H is a normal subgroup the coset relation becomes a congruence relation.

Cosets have important applications in computer science, e.g. in the development of efficient codes needed in the transmission of information and in the design of fast adders.

coset See CROSS-STRATIFICATION.