Definition:Coset/Left Coset

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Let $\struct {S, \circ}$ be an algebraic structure.

Let $\struct {H, \circ}$ be a subgroup of $\struct {S, \circ}$.

The left coset of $x$ modulo $H$, or left coset of $H$ by $x$, is:

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

That is, it is the subset product with singleton:

$x \circ H = \set x \circ H$

Also defined as

It is usual for the algebraic structure $S$ in fact to be a group.

Hence, when $\struct {S, \circ}$ is a group, the left coset of $H$ by $x$ is the equivalence class of $x$ defined by left congruence modulo $H$.

Some sources (see P.M. Cohn: Algebra Volume 1 (2nd ed.), for example) order the operands in the opposite direction, and hence $H \circ x$ is a left coset.

Also see

  • Results about cosets can be found here.