Action of Smooth Group on Set

The action of smooth groups on the set for revealing structure of smooth groups has not been studied. In order to solve the problem, this paper studies three actions of smooth groups on sets: 1. The left translation action on set. 2. The left translation action on the set consisted of all cosets w.r.t. smooth subgroup H. and 3. The conjugate action. This paper proves four theorems. 1. The first action induces smooth homomorphism. 2. Cayley theorem, that is, smooth group is isomorphic with some smooth permutation group. 3. The second action induces a smooth homomorphism whose kernel is in H. 4. The third action induces a smooth automorphism whose kernel consists of commutative elements with all elements in smooth group. This paper enriches the structure of smooth groups.