Center of a Group
Definition: The set of all those elements of a group
which commute with every element of
is called the center of the group
. Symbolically
Theorem: The center 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
.