Question

(a) Find the order of groups ${\mathit{U}}_{10}$, ${\mathit{U}}_{12}$and ${\mathit{U}}_{24}$.

Short answer

The order group ${\mathit{U}}_{10}$is 6, order of group ${\mathit{U}}_{12}$ is 4 and order of group ${\mathit{U}}_{24}$ is 8.

Step-by-step solution

To find t

The definition of the order of group states that,

Let Gbe a group. Then G is said to be finite group (or of finite order) if it has a finite number of elements. It is also called as cardinality of the group.

Here, the number of elements in group Gis called order ofG .It is denoted by o(G).

A group having infinitely many elements is said to be of infinite order.

Now, in order to find the order of ${\mathit{U}}_{10}$, ${\mathit{U}}_{12}$and ${\mathit{U}}_{24}$, let us see the definition of ${\mathit{U}}_n$ group, that is, unit modulo n group,

${\mathit{U}}_{\mathbf{n}}=\left\{x\in N/1<x\le n;gcd\left(x,n\right)=1\right\}$

role="math" localid="1653451937338" $\left({\mathit{U}}_n,{\mathit{x}}_n\right)$is group under multiplication modulo n . Moreover, it is an abelian group under multiplication.

We have,

$U_{10}=\left\{1,3,6,8,9\right\}$

Here, the number of elements in the group ${\mathit{U}}_{10}$are 6.

Hence, $o\left(U_{10}\right)=6$.

To find the order of group U24

We have,

$U_{12}=\left\{1,5,7,11\right\}$

Here, the number of elements in the group ${\mathit{U}}_{12}$are 4.

Hence,$o\left(U_{12}\right)=4$.

To find the order of group U24 

We have,

$U_{24}=\left\{1,5,7,11,13,17,19,23\right\}$

Here, the number of elements in the group ${\mathit{U}}_{24}$are 8.

Hence, $o\left(U_{24}\right)=8$.