Question

If $\mathit{n}\mathbf{ }\mathbf{>}\mathbf{ }\mathbf{2}$, prove that the order of the group ${\mathit{U}}_{\mathbf{n}}$is even.

Short answer

We proved that, order of group $U_n$is even if $n>2$.

Step-by-step solution

To find order of subgroup

Suppose $U_n$is a group of units which is a group of integers under multiplication modulo $n$.

Let$n>2$ .

Suppose the subset $\left\{1, -1\right\}$is a subgroup of $U_n$.

Now, ${\left(-1\right)}^2=1 , 1\left(-1\right)=\left(-1\right)1=1$.

This implies that the order of subgroup is 2.

To find order of group  Un

From Lagrange’s Theorem,

Order of subgroup $\left\{1, -1\right\}$divides order of group $U_n$.

This implies that order of $U_n$is even.

Hence, order of group $U_n$is even if $n>2$.