# Centre 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 centre of the group $G$. Symbolically

Theorem: The centre $Z$ 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 $x \in G$

Now

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

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

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