# coset

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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: xH = {xh | hH} Hx = {hx | hH}

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 xH = Hx

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 g1H = g2H

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.

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