groups

Question

Explain : In a group (Z,+) the order of any element except zero does not exist.

Answer 1

Order/period of a group is the smallest positive integer m such that a^m =e where e is the identity element and a is an element of the group.In the given group (Z,+), the identity element is 0 since a+0=a.

We have to find a^m=0.It is not possible to satisfy this condition except for a=0 and m=1.Hence the order of any element except 0 doesn't exist in a group( Z, +)