Find all elements of finite order in each of the following g

Find all elements of finite order in each of the following groups. Here the \"*\" indicates the set with zero removed. Z Q^* R^*

Solution

sol. (A) For the additive group Z of integers, every non-zero element has infinite order. (Of course, here, we use additive notation, so to calculate the order of gZ, we are looking for the least positive integer n such that ng=0, if any. But, unless g=0, there is no such n, so the order of g is .)

(B) let Q* be the group of non zero rational numbers under multiplication. since 1 is the identity of Q* then by definition of finite order of an element ,there exist x belongs to Q* such that x^n =1. this equation is satisfied by only x=1 and x=-1.

thus the only two elements of Q* of finite order are 1 and -1.

ord(1)=1 and ord(-1) =2

(C) similarly R* has only two elements of finite order which are 1 and -1