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

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

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

 \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.