# Coset in a group

$G$ by a subgroup $H$ (from the left)

A set of elements of $G$ of the form $$aH = \{ ah : h \in H \}$$ where $a$ is some fixed element of $G$. This coset is also called the left coset by $H$ in $G$ defined by $a$. Every left coset is determined by any of its elements. $aH = H$ if and only if $a \in H$. For all $a,b \in G$ the cosets $aH$ and $bH$ are either equal or disjoint. Thus, $G$ decomposes into pairwise disjoint left cosets by $H$; this decomposition is called the left decomposition of $G$ with respect to $H$. Similarly one defines right cosets (as sets $Ha$, $a \in G$) and also the right decomposition of $G$ with respect to $H$. These decompositions consist of the same number of cosets (in the infinite case, their cardinalities are equal). This number (cardinality) is called the index of the subgroup $H$ in $G$. For normal subgroups, the left and right decompositions coincide, and in this case one simply speaks of the decomposition of a group with respect to a normal subgroup.

The (left) cosets are the equivalence classes for the equivalence relation $a \sim b \Leftrightarrow a^{-1}b \in H$.