Let \(H\subseteq G\) be a subgroup of the group \(G\) and \(g\in G\). The left coset of \(H\) determined by \(g\) is the set
\[ gH=\{gh\mid h\in H\}. \]The right coset \(Hg\) is defined similarly.
The set of left cosets of \(H\) in \(G\) is denoted by \(G/H\). We call the cardinality of \(G/H\) index of \(H\) in \(G\).
Remarks
- The equivalence classes of congruence module \(H\) are the left cosets of \(H\).
- \(G/H\) is the partition of the equivalence relation congruence modulo \(H\).