Let \((G,\ast)\) be a group and \(H\subseteq G\) its subgroup. Then the following properties hold for cosets:
(1) If \(|H|=n\), then \(|aH|=n\) (respectively \(|Ha|=n\)), i.e. cosets of a finite subgroup have the same order as this subgroup.
(2) The relation \(a\sim_l b:=a\in bH\) (respectively \(a\sim_r b:=a\in Hb\)) defines an equivalence relation on the group \(G\). In particular, it partitions the group \(G\) into left cosets (respectively right cosets). The quotient set of these left cosets (respectively right cosets) is denoted by \(G/\sim_l=\{aH: a\in G\}\) (or respectively \(G/\sim_r=\{Ha: a\in G\}\)).
Proofs: 1
