Question

If N is a subgroup of$\mathit{Z}\left(G\right)$ , prove that N is a normal subgroup of G .

Short answer

It is proved that N is a normal subgroup of G

Step-by-step solution

Center of a group

Let G be a group. The center $Z\left(G\right)$ of a group G is defined as

$Z\left(G\right)=\left\{a\in G/ax=xa\forall x\in G\right\}$

Now, suppose N is subgroup of$Z\left(G\right)$

Let, $a,b\in N\Rightarrow ab=ba\in N$ [ N is subgroup of$Z\left(G\right)$ ]

We have to prove N is a normal subgroup of G

Normal subgroup

A subgroup H of group G is said to be normal subgroup of G if

$x\in G$and $h\in H\Rightarrow xh{x}^{-1}\in H$

Here, we have

$a,b\in N\Rightarrow ab=ba\in N$ [ N is subgroup of $Z\left(G\right)$]

$\Rightarrow ab{a}^{-1}=ba{a}^{-1}$ [ is subgroup. multiplying from right side]

$$\begin{gathered} \Rightarrow ab{a}^{-1}=b \\ \Rightarrow ab{a}^{-1}\in N \end{gathered}$$

Therefore, N is a normal subgroup of G

Hence proved.