Definition:Coset/Right Coset

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Definition

Let $G$ be a group, and let $H \le G$.

The right coset of $y$ modulo $H$, or right coset of $H$ by $y$, is:

$H y = \set {x \in G: \exists h \in H: x = h y}$

This is the equivalence class defined by right congruence modulo $H$.


That is, it is the subset product with singleton:

$H y = H \set y$


Also defined as

The definition given here is the usual one, but some sources (see 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.), for example) order the operands in the opposite direction, and hence $x H$ is a right coset.


Also see

  • Results about cosets can be found here.


Sources