Question

If $n>2$, prove that the order of the group $U_n$ is even.

Short answer

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

Step-by-step solution

To find the order of the 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 $\{1,-1\}$is a subgroup of $U_n$.

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

This implies that the order of the subgroup is 2.

To find the order of group Un

From Lagrange’s Theorem,

The order of subgroup $\{1,-1\}$divides the order of group $U_n$.

This implies that the order of $U_n$is even.

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