So far, we have always considered transformation groups and the sets that are acted upon in pairs. In Sect. 3.1, we focus on algebraic structures of transformation groups leaving aside the sets acted upon. We introduce an abstract framework of group theory. In Sect. 3.2 , we introduce the concept of subgroups. In Sect. 3.3, we study the actions of groups on themselves, where the concept of coset decomposition or orbit decomposition plays the central role.