The Normalizer/Centralizer Theorem.

“Beauty is so many things… and you are in all of them,” I said. “Wow! I thought that you were as romantic as a brick to the head or a nuclear bomb’s guts”, Susanne replied, Apocalypse, Anawim, #justothepoint.

“When bias narratives, misleading news without context, and blatant lies spread like wildfire, history needs to be rewritten and truth needs to be cancelled”, Anawim, #justothepoint.

First Isomorphism Theorem. Let Φ be a group homomorphism from G to G’. Then, the function or mapping from G/Ker(Φ) to Φ(G), defined by gKer(Φ) → Φ(g) is an isomorphism, i.e., G/Ker(Φ) ≋ Φ(G). In particular, if Φ is onto, G/Ker(Φ) ≋ G’.

More examples

• Let Φ: ℂ* → ℝ⁺ -the group of positive real numbers under multiplication-, the map defined by a + bi → |a + bi| = $\sqrt{a^2+b^2}$. Φ is an homomorphism, onto, and Ker(Φ) = {z ∈ ℂ*: |z| = 1}. Therefore, ℂ*/Ker(Φ) = ℂ*/S¹ ≋ ℝ⁺.

• Consider a group homomorphism Φ: (Q₈, x) → ℤ₂ x ℤ₂ where Q₈ is the Hamilton Quaternion group, defined by Φ(i) = (1, 0) and Φ(j) = (0, 1).

Φ(k) =[k = ij] Φ(i)Φ(j) = (1, 0) + (1, 0) = (1, 1). Φ(-k) = -Φ(k) = -(1, 1) = (-1, -1) =_{ℤ2 x ℤ2} (1, 1)
Φ(-i) = -Φ(i) = -(1, 0) = (-1, 0) =_{ℤ2 x ℤ2} (1, 0)
Φ(-j) = -Φ(j) = -(0, 1) = (0, -1) =_{ℤ2 x ℤ2} (0, 1)
Φ(-1) =[i² = -1] = Φ(i) + Φ(i) = (1, 0) + (1, 0) =_{ℤ2 x ℤ2} (0, 0). Φ(1) = -Φ(1) = -(0, 0) = (0, 0).

Therefore, Φ is onto, Ker(Φ) = {1, -1} = ⟨-1⟩, and Q₈/⟨-1⟩ ≋ ℤ₂ x ℤ₂.

• Let S¹ be the unit circle in ℂ with multiplication as the group operation, S¹ = {z | |z| = 1}. Let Φ: ℝ → S¹, Φ(r) = e^i2πr is a homomorphism, onto, and Ker(Φ) = ℤ ⇒ ℝ/ℤ ≋ S¹.

Recall

Definitions. Two subgroups H and K of a group G are named conjugate in G if there is an element g ∈ G such that H = gKg^-1. gAg^-1 = {gag^-1 | a ∈ A}. The normalizer of H in the group G is defined as N_G(H) = N(H) = {g ∈ G: gHg^-1 = H} = {g ∈ G: gH = Hg}. Clearly C_G(H) ⊆ N_G(H) and both are subgroups of G.

Let G be a group, and H let be a non-empty subset of G.The centralizer of a group G is the subset of elements of G that commute with every element of H. Formally, C_G(H) = {x ∈ G | xhx^-1 = h ∀h ∈ H} = {x ∈ G | xh = hx ∀ h ∈ H}.

Let (A, *) and (B, ⋄) be two binary algebraic structures. A homomorphism is a structure-preserving map between two algebraic structures of the same type (groups, rings, fields, vector spaces, etc.) or, in other words, a map ϕ: A → B, such that ∀x, y ∈ S : ϕ(x ∗ y) = ϕ(x) ⋄ ϕ(y).

Notice that ϕ may not be one to one (injection), nor onto (surjection). An isomorphism is a bijective homomorphism, i.e., one-to-one and onto. In other words, let (S, *) and (S’, ⋄) be two binary algebraic structures of the same type. An isomorphism of S with S' is a 1-1 function ϕ mapping from S onto S′ such that the homomorphism property holds: ∀x, y ∈ S : ϕ(x ∗ y) = ϕ(x) ⋄ ϕ(y). S and S’ are said to be isomorphic binary structures and we denote or write it by S ≃ S'.

An automorphism is an isomorphism from a group to itself. Let G be a group, and let a be a fixed or given element of G, a ∈ G.An inner automorphism of G (induced or given by a) is defined by the conjugation action of the fixed element a, called the conjugating element i.e., Φ_a: G → G defined by Φ_a(x) = a·x·a^-1 ∀x ∈ G.

The Normalizer/Centralizer Theorem. Let H be a subgroup of G. Consider the mapping from the normalizer of H in G, N(H) to Aut(H) given by γ: g → Φ_g, where Φ_g is the inner automorphism induced or given by g, that is, Φ_g(h) = ghg^-1 ∀h ∈ H. This mapping is indeed a homomorphism with kernel the centralizer of H in G. The centralizer is a normal subgroup of the normalizer and N(H)/C(H) is isomorphic to a subgroup of Aut(H).

Proof.

Let γ be the mapping γ: (N(H),·) → (Aut(H),∘), defined by x → Φ_x where Φ_x(h) = xhx^-1 ∀h ∈ H.

• The mapping is indeed a homomorphism. ∀x, y ∈ N(H), γ(x)γ(y) = Φ_x°Φ_y

Φ_x°Φ_y ⇒ [° represents composition] ∀h ∈ H: Φ_x°Φ_y(h) = Φ_x(Φ_y(h)) =[By definition of inner automorphism] Φ_x(yhy^-1) =[By definition of inner automorphism] x(yhy^-1)x^-1 =[Associative] (xy)h(y^-1x^-1) =[The Socks and Shoes Principle] (xy)h(xy)^-1 = Φ_xy(h)

• It kernel is the centralizer of H in G. x ∈ Ker(γ) ↭ γ(x) = Φ_x = id_H, the identity of Aut(H) ↭ ∀h ∈ H: Φ_x(h) = xhx^-1 = id_H(h) = h ↭ ∀h ∈ H, xhx^-1 = h ↭ ∀h ∈ H, xh = hx ↭ x ∈ C(H) ↭ Ker(γ) = C(H).

  Recall: C_G(H) or simply, C(H) = {x ∈ G | xhx^-1 = h ∀h ∈ H} = {x ∈ G | xh = hx ∀ h ∈ H}.

• Recall: If K is a normal subgroup of G', then Φ^-1(K) = {k ∈ G: Φ(k) ∈ K} is a normal subgroup of G. In particular, the trivial subgroup is obviously normal, {e’} ◁ G’, then Φ^-1({e’}) = Ker(Φ) ◁ G. In words, kernels are normal subgroups of the domains, and therefore C_G(H) ◁ N(H). In words, the centralizer is a normal subgroup of the normalizer.

• By the First Isomorphism Theorem for Groups: N(H)/C(H) is isomorphic to Img(Φ). Futhermore, if Φ homomorphism Φ: G → G’ ⇒ Img(Φ) ≤ G’. Therefore, Img(Φ) ≤ Aut(H), and N(H)/C(H) is isomorphic to a subgroup of Aut(H).

💡 We have already demonstrated that the kernel of a group homomorphism is normal, but is the inverse also true?

Theorem. Every normal subgroup of a group G, N ◁ G, is the kernel of a homomorphism of G.In particular, a normal subgroup N is the kernel of the mapping γ from G to G/N defined by g → gN. Therefore, a subgroup is normal if and only if it is the kernel of a homomorphism.

Proof.

Let’s define a mapping γ: G → G/N, γ(g) = gN.

• is it well-defined? If x and y are in the same coset of N, say gN, x, y ∈ gN ⇒ [A left (or right) coset is uniquely determined or represented by any of its elements, i.e., aH = bH iff a ∈ bH] xN = yN = gN ⇒ γ(x) = γ(y) = γ(g).
• is γ homomorphism? γ(g)γ(h) = gNhN = [By assumption, N is normal in G, N ◁ G, the multiplication of any two left (or right) cosets gN and hN yields the left coset (gh)N] ghN = γ(gh)
• What is it kernel?

1. N ⊆ Ker(γ)? ∀k ∈ N, γ(k) = kN [A coset H absorbs an element if and only if the element belongs to it, aH = H iff a ∈ H] = N (the identity of the factor group) ⇒ k ∈ Ker(γ).
2. Conversely, Ker(γ) ⊆ N? If γ(k) = kN = N ⇒ [A coset H absorbs an element if and only if the element belongs to it, aH = H iff a ∈ H] k ∈ N ⇒[1, 2] Ker(γ) = N ∎

Bibliography

1. NPTEL-NOC IITM, Introduction to Galois Theory.
2. Algebra, Second Edition, by Michael Artin.
3. LibreTexts, Calculus. Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
4. Field and Galois Theory, by Patrick Morandi. Springer.
5. Michael Penn, Andrew Misseldine, and MathMajor, YouTube’s channels.
6. Contemporary Abstract Algebra, Joseph, A. Gallian.
7. MIT OpenCourseWare, 18.01 Single Variable Calculus, Fall 2007 and 18.02 Multivariable Calculus, Fall 2007, YouTube.
8. Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences.