Category:Definitions/Cosets

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This category contains definitions related to Cosets.
Related results can be found in Category:Cosets.

Left Coset

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

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

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


That is, it is the subset product with singleton:

$x H = \set x H$


Right Coset

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:

$H y = H \set y$