Question

10. Find the order of every element in each group.

(e)$\mathrm{\mathbb{Z}}$

Short answer

The order of element 0 in$\mathrm{\mathbb{Z}}$ is 1 and$x\ne 0$ in$\mathrm{\mathbb{Z}}$ is infinite.

Step-by-step solution

Order of an element

The order of an element x in a group G is the smallest positive integer n such that $x^n=e$. The order of the element x is denoted by $\left|x\right|$.

Order of elements in ℤ

Here the group$\mathrm{\mathbb{Z}}$has an infinite number of elements.

By the definition the order of identity element is 1 and for any other element x in $\mathrm{\mathbb{Z}}$there does not exists any $n>0$such that $x^n=e$.

The order of each element in $\mathrm{\mathbb{Z}}$is given in the below table:

Element	Order
0	1
$x\ne 0$	$\infty$

Therefore, the order of element 0 in$\mathrm{\mathbb{Z}}$ is 1 and$x\ne 0$ in$\mathrm{\mathbb{Z}}$ is infinite.