# Centre of a Group

__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 .

__Proof__**:**

We have . First we shall prove that is a subgroup of .

Let , then and for all

We have , for all

Now

Thus,

Therefore, is a subgroup of .

Now, we shall show that is a normal subgroup of . Let and , then

Thus, ,

Therefore, is a normal subgroup of .