The center of a group G is a subset containing those elements of G that commute with every element of the group G. In this article, we will learn about the center of a group and show that it is a normal subgroup.

Table of Contents

## Center of a Group Definition

Let (G, o) be a group. Then the set defined below

Z(G) = {g ∈ G: ga=ag ∀ a∈G}

is called the center of G. The elements of Z(G) are called the *central elements* of G.

## Center of a Group Example

(1) If G is an abelian group, that is, a commutative group, then every element commutes with all elements of G. Thus, by definition, the center of G will be the group G itself. That is,

Z(G) = G if G is abelian.

(2) We know that (R, +) is an abelian group where R denotes the set of all real numbers. By the first example, the center of R is R itself. In other words,

Z(R) = R.

## Center of a Group is a Subgroup

Let (G, o) be a group with center Z(G) = {g ∈ G: ga=ag ∀ a∈G}. Then Prove that Z(G) is a subgroup of G.

*Proof:*

Let e_{G} denote the identity in G. Note that e_{G}a=ae_{G} ∀ a∈G. Thus, we obtain that e_{G} ∈ Z(G). This shows that Z(G) is non-empty.

Let x, y ∈ Z(G).

We will show that xy ∈ Z(G). As x, y ∈ Z(G), we have that

xa=ax ∀ a∈G …(I)

ya=ay ∀ a∈G …(II)

Now, for all a∈G we have

(xy)a = x(ya)

=x(ay) by (II)

= (xa)y = (ax)y by (I)

⇒ (xy)a=a(xy) a∈G. This shows that xy∈Z(G).

Now we will show that x^{-1} ∈ Z(G). By the relation (I), for all a∈G we have that

x^{-1}(xa)x^{-1}=x^{-1}(ax)x^{-1}

⇒ (x^{-1}x)ax^{-1}=(x^{-1}a)xx^{-1}

⇒ ax^{-1}=x^{-1}a as we have x^{-1}x=xx^{-1}=e_{G}.

⇒ x^{-1} ∈ Z(G).

Thus, we have shown that

- xy∈Z(G) for all x, y ∈ Z(G).
- x
^{-1}∈ Z(G) for all x ∈ Z(G).

Therefore, Z(G), the center of the group G, is a subgroup of G.

## Center of a Group is Abelian

By definition of the center of a group G, we have that if g∈Z(G) then ga=ag ∀ a∈G. In particular, ga=ag ∀ a∈Z(G). This shows that elements of Z(G) commute with every element of Z(G). Hence, Z(G) is abelian.

As Z(G) is a subgroup, we conclude that Z(G), the center of the group G, is an abelian group or a commutative group.

## Center of a Group is a Normal Subgroup

Prove that the center Z(G) is a normal subgroup of G.

*Proof:*

From the above, we have that the center Z(G) is a subgroup of G. Now, we will show that it is a normal subgroup of G.

Let x∈G and g∈Z(G). As Z(G) is a subgroup, we have g^{-1} ∈Z(G).

Therefore, xgx^{-1} = x(gx^{-1}) = x(x^{-1}g) = g ∈Z(G).

That is, xgx^{-1} ∈Z(G) ∀ x∈G and ∀ g∈Z(G).

Hence, the center Z(G) of the group G is a normal subgroup of G.

**Related Topics:**

**Group Theory: Definition, Examples, Orders, Types, Properties, Applications**