Question

Give an example of a group $\mathit{G}$such that $\mathit{G}\mathbf{/}\mathit{Z}\mathbf{\left(}\mathbf{G}\mathbf{\right)}$ is nonabelian.

Short answer

The group $S_3$ with the quotient group$S_3/Z\left(S_3\right)$ is nonabelian.

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 $N$.

Group S3 with  S3/Z(S3) is nonabelian

Consider the group $S_3$, where the center of $S_3$ is the identity that is $Z\left(S_3\right)=\left\{e\right\}$.

Then the quotient group $S_3/Z\left(S_3\right)$ is the group $S_3$ itself, which is nonabelian.

Therefore, the group $S_3$ with the quotient group $S_3/Z\left(S_3\right)$ is nonabelian.