coset
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 g_{1} and g_{2} of G as follows: g_{1} ρ g_{2} iff g_{1} ◦ H = g_{2} ◦ 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.
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"coset." A Dictionary of Computing. . Encyclopedia.com. 22 Oct. 2017 <http://www.encyclopedia.com>.
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coset
coset See CROSSSTRATIFICATION.
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"coset." A Dictionary of Earth Sciences. . Encyclopedia.com. 22 Oct. 2017 <http://www.encyclopedia.com>.
"coset." A Dictionary of Earth Sciences. . Encyclopedia.com. (October 22, 2017). http://www.encyclopedia.com/science/dictionariesthesaurusespicturesandpressreleases/coset
"coset." A Dictionary of Earth Sciences. . Retrieved October 22, 2017 from Encyclopedia.com: http://www.encyclopedia.com/science/dictionariesthesaurusespicturesandpressreleases/coset