Center of a Group

Definition: The set $Z$ of all those elements of a group $G$ which commute with every element of $G$ is called the center of the group $G$. Symbolically

$$Z = \left\{ {z \in G:zx = xz \Rightarrow x \in G} \right\}$$

Theorem: The center $Z$ of a group $G$ is a normal subgroup of $G$.

Proof:
We have $Z = \left\{ {z \in G:zx = xz\,\,\,\forall x \in G} \right\}$. First we shall prove that $Z$ is a subgroup of $G$.

Let ${z_1},{z_2} \in Z$, then ${z_1}x = x{z_1}$ and ${z_2}x = x{z_2}$ for all $x \in G$

We have ${z_2}x = x{z_2}$, for all $x \in G$

$$\Rightarrow {z_2}^{ - 1}\left( {{z_2}x} \right){z_2}^{ - 1} = {z_2}^{ - 1}\left( {x{z_2}} \right){z_2}^{ - 1}$$

$$\Rightarrow x{z_2}^{ - 1} = {z_2}^{ - 1}x\,\,\,\forall x \in G$$

$$\Rightarrow {z_2}^{ - 1} \in Z$$

Now

$$\begin{gathered} \left( {{z_1}{z_2}^{ - 1}} \right)x = {z_1}\left( {{z_2}^{ - 1}x} \right) = {z_1}\left( {x{z_2}^{ - 1}} \right) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {{z_1}x} \right){z_2}^{ - 1} = \left( {x{z_1}} \right){z_2}^{ - 1} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = x\left( {{z_1}{z_2}^{ - 1}} \right) \\ \Rightarrow {z_1}{z_2}^{ - 1} \in Z \\ \end{gathered}$$

Thus, ${z_1},{z_2} \in Z \Rightarrow {z_1}{z_2}^{ - 1} \in Z$

Therefore, $Z$ is a subgroup of $G$.

Now, we shall show that $Z$ is a normal subgroup of $G$. Let $x \in G$ and $z \in Z$, then

$$\begin{gathered} xz{x^{ - 1}} = \left( {xz} \right){x^{ - 1}} = \left( {zx} \right){x^{ - 1}} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, = z \in Z \\ \end{gathered}$$

Thus, $x \in G$, $z \in Z \Rightarrow xz{x^{ - 1}} \in Z$

Therefore, $Z$ is a normal subgroup of $G$.