Centre of a group

The Centre of a group G is denoted by Z(G) or C(G) or Z and is defined as
             Z = C(G) = Z(G) = {g ∈ G | gx = xg, ∀ x ∈ G}.
The centre of a subgroup of G, which by definition is abelian (i.e., commutative). As a subgroup, it is always normal, and indeed characteristic, but it need not be fully characteristic. The quotient group G/Z(G) is isomorphic to the group of inner automorphisms of G.
A group G is abelian if and only if Z(G) = G. At the other extreme, a group is said to be centerless if Z(G) is trivial, 
i.e., consists only of the identity element.
The elements of the centre are sometimes called central.

Examples:

• The centre of an abelian group G is all of G.
• The centre of a non-abelian simple group is trivial.
• The centre of the dihedral group D_n is trivial when n is odd. When n is even, the centre consists of the identity element together with the 180 degree rotation of the polygon.
• The centre of the quaternion group Q₈ = {1, -1, i, -i, j, -j, k, -k} is {1, -1}.
• The centre of the symmetric group S_n is trivial for n ≥ 3.
• The centre of the alternating group A_n is trivial for n ≥ 4.
• The centre of the orthogonal group O(n, F) is {I_n, -I_n}.
• The centre of the multiplicative group of non-zero quaternions is the multiplicative group of non-zero real numbers.
• The quotient group G/Z(G) is not isomorphic to the quaternion group Q₈.

Theorem: The centre Z(G) of a group G is a subgroup of G.

Proof: Let Z(G) = {g ∈ G | gx = xg, ∀ x ∈ G} be the centre of a group G.
Clearly Z(G) ⊆ G.
Since ex = xe, ∀ x ∈ G
∴ e ∈ Z(G).
∴ Z(G) is a non-empty subset of G.
Let g₁ , g₂ ∈ Z(G) be any two elements, then
           g₁x = xg₁       and   g₂x = xg₂ ,   ∀ x ∈ G
⇒       xg₂^-1 = g₂^-1x.
Now, x(g₁ g₂^-1) = (xg₁) g₂^-1= (g₁x)g₂^-1
                           = g₁ (xg₂^-1) = g₁ (g₂^-1x)
                           = (g₁ g₂^-1)x
i.e.,   x(g₁ g₂^-1)  =  (g₁ g₂^-1)x,      ∀ x ∈ G
So, g₁ g₂^-1 ∈ Z(G),      ∀ g₁ , g₂ ∈ Z(G).
Hence, Z(G) is a subgroup of G.

Theorem: G is abelian group iff Z(G) = G.

Proof: Firstly, let Z(G) = G
i.e., Z (G) = {g ∈ G | gx = xg, ∀ x ∈ G} = G
⇒   xy = yx,         ∀ x, y ∈ G
⇒ G is an abelian group.

Conversely: Let G be abelian.
⇒   xy = yx,         ∀ x, y ∈ G.
To show that Z(G) = G.
Since Z(G) is a subgroup of G
∴ Z(G) ⊆ G
Now, let x ∈ G be any element
∵ G is abelian
∴ xy = yx,         ∀ y ∈ G.
⇒ x ∈ Z(G)
∴ G ⊆ Z(G)
Hence Z(G) = G.

Theorem: Let G be a group, then Z(G) ≤ G. 
Moreover, G/Z(G) ≅ Inn(G) ≤ Aut(G) (where Inn(G) is the set of inner automorphism).
Proof: We know that Φ : G → Aut(G) : g → i_g is a homomorphism.
Note that 
                ker Φ = {g ∈ G : i_g = idG}
                           = {g ∈ G : g^-1hg = h for all h ∈ G}
                           = {g ∈ G : hg = gh for all h ∈ G}
                           = Z(G)
from where the conclusion follows from our earlier characterization of normality. 
the fact that  G/Z(G) ≅ Inn(G) now follows immediately from the First Isomorphism Theorem.