Let \((G,\ast)\) be a group, and \(H\subseteq G\) its subgroup and \(a\in G\) any element of \(G\). Then the set \[aH:=\{a\ast h:h\in H\}\] is called the left coset of \(H\) with respect to \(a\) and \[Ha:=\{h\ast a:h\in H\}\] is called the right coset of \(H\) with respect to \(a\).
Definitions: 1
Lemmas: 2 3
Proofs: 4 5 6 7 8 9
Theorems: 10