Definition: The set of all those elements of a group which commute with every element of is called the centre of the group . Symbolically
Theorem: The centre of a group is a normal subgroup of .
We have . First we shall prove that is a subgroup of .
Let , then and for all
We have , for all
Therefore, is a subgroup of .
Now, we shall show that is a normal subgroup of . Let and , then
Therefore, is a normal subgroup of .