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 eG denote the identity in G. Note that eGa=aeG ∀ a∈G. Thus, we obtain that eG ∈ 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-1x)ax-1=(x-1a)xx-1
⇒ ax-1=x-1a as we have x-1x=xx-1=eG.
⇒ 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-1g) = 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
