Coset

Consider a countable Subgroup $H$ with Elements $h_i$ and an element $x$ not in $H$, then

$$x h_i$$ (1)

$$h_i x$$ (2)

for $i=1$, 2, ... are left and right cosets of the Subgroup $H$ with respect to $x$. The coset of a Subgroup has the same number of Elements as the Subgroup. The Order of any Subgroup is a divisor of the Order of the Group. The original Group can be represented by

$$G = H+x_1 H+x_2 H+\ldots.$$ (3)

For $G$ a not necessarily Finite Group with $H$ a Subgroup of $G$, define an Equivalence Relation $x\sim y$ if $x = hy$ for some $h$ in $H$. Then the Equivalence Classes are the left (or right, depending on convention) cosets of $H$ in $G$, namely the sets

$$\{x \in G: x = ha {\rm\ for\ some\ } h {\rm\ in\ } H\},$$ (4)

where $a$ is an element of $G$.

See also Equivalence Class, Group, Subgroup