Let and for let be the fractional part of , i.e., Prove that is a well-defined binary operation on and that is an abelian group under .
Solution: Before we begin, note that for all and as follows. For all integers , we have iff iff iff .
We first show that is well defined. Suppose . There are two cases for . If , then we have and so . If , we have . But since , . Thus . So is indeed a binary operator on .
We now show that is associative. To that end, let . Then we have the following. We now show that 0 is the identity element of under . If , then . Thus we have and We now show that every element has an inverse. If , then . Then and similarly Certainly, if then . Thus is a group under . It is also clear that . Hence is an abelian group under .