Abstract

• We present that a mapfrom a nonempty set into itself where each is in a group ; namely, is called an action of on . We next present the Orbit-Stabilizer Theorem: If acts on ,  then the order of orbit of is equal to the index of stabilizer of in . Last, we show that if an infinite group has a subgroup of finite index then it also has a normal infinite subgroup of finite index, which is the consequence of this theorem.