Question

Complete the table in example 2 and verify that every nonidentity element of $D_4/M$of order 2.

Short answer

The order of every nonidentity element in $D_4/M$ is 2.

Step-by-step solution

Normal subgroup and Quotient group

Let $N$ be the normal subgroup of $G$. Then

1. $G/N$is a group under the operation defined by $\left(Na\right)\left(Nc\right)=Nac$.

2. If $G$is finite, then the order of $G/N$ is $\left|G\right|/\left|N\right|$.

3. If $G$ is an abelian group, then so is $G/N$.

The group $G/N$ is called the quotient group or factor group of $G$ by$N$ .

Complete table of D4/M

The elements in the group $D_4/M$are $M{r}_0,M{r}_1,Mh,Md$. The complete table of $D_4/M$is shown below:

Nonidentity element of D4/M is of order 2

The identity element of $D_4/M$ is $M{r}_0$. From the table, it is clear that $\left(M{r}_1\right)\left(M{r}_1\right)=\left(M{r}_0\right)$, $\left(Mh\right)\left(Mh\right)=\left(M{r}_0\right)$, $\left(Md\right)\left(Md\right)=\left(M{r}_0\right)$.

Therefore, the order of every nonidentity element in $D_4/M$ is 2.