views updated Jun 11 2018

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.


views updated May 08 2018