In this note, the concepts of (G 1 , G 2)-open, (G 1 , G 2)-closed, (G 1 , G 2)-quotient and (G 1 , G 2)–perfect mappings on arbitrary sets are introduced and some theorems on them are established firstly. In particular, some results improve the corresponding results in [17]. Secondly, we give a partial answer to the question posed by L. Liu [14]. Finally, some properties of G-topological groups, G-connectedness and totally G-disconnectedness in G-topological groups are discussed