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 $$G$$.