Question

Give an example of a group that contains non-identity elements of finite order and of infinite order.

Short answer

The order of non-identity element 2 is 4, order of 3 is 4 and order of 4 is 2.

The group containing non-identity elements of infinite order is $\left(\mathrm{\mathbb{Z}},+\right)$.

Step-by-step solution

To find a group containing non-identity elements of finite order

Let$G=\left\{1,2,3,4\right\}$be a group under multiplication modulo 5

Now, let us find the order of elements of group G.

$o\left(1\right)=1^1=1$

$$\begin{gathered} o\left(2\right): \\ {2}^1=2\ne 1 \\ {2}^2=2{\times }_{5}2=4\ne 1 \\ {2}^3=2{\times }_{5}2{\times }_{5}2{\times }_{5}2=1 \\ \therefore o\left(2\right)=4 \end{gathered}$$

$$\begin{gathered} o\left(3\right): \\ {3}^1=3\ne 1 \\ {3}^2=3{\times }_{5}3=4\ne 1 \\ {3}^3=3{\times }_{5}3{\times }_{5}3=2\ne 1 \\ {3}^4=3{\times }_{5}3{\times }_{5}3{\times }_{5}3=1 \\ \therefore o\left(3\right)=4 \end{gathered}$$

$$\begin{gathered} o\left(4\right): \\ {4}^1=4\ne 1 \\ {4}^2=4{\times }_{5}4=1 \\ \therefore o\left(4\right)=2 \end{gathered}$$

Hence, the order of non-identity elements 2, 3, and 4 are 4, 4 and 2 respectively.

It means that the orders of these elements are finite.

To find a group containing non-identity elements of infinite order

Consider a group $\left(\mathrm{\mathbb{Z}},+\right)$, an abelian group with respect to usual addition of integers.

Let $a\in \mathrm{\mathbb{Z}},a\ne 0$, be any non-identity element.

Assume that $o\left(a\right)=k<\infty$.Then,

$\underset{ktimes}{\underbrace{a+a+a+\dots +a}}=ka=0$and $k\ge 2$.

$ka=0$and$a\ne 0$ implies that $k=0$, which is a contradiction since $k\ge 2$.

This contradiction shows that the order of is infinite.

Therefore,$\left(\mathrm{\mathbb{Z}},+\right)$ is an example of group containing non-identity elements of infinite order.