# The Normalizer/Centralizer Theorem.

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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) =2 x ℤ2 (1, 1)
Φ(-i) = -Φ(i) = -(1, 0) = (-1, 0) =2 x ℤ2 (1, 0)
Φ(-j) = -Φ(j) = -(0, 1) = (0, -1) =2 x ℤ2 (0, 1)
Φ(-1) =[i2 = -1] = Φ(i) + Φ(i) = (1, 0) + (1, 0) =2 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) = ei2πr is a homomorphism, onto, and Ker(Φ) = ℤℝ/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 NG(H) = N(H) = {g ∈ G: gHg-1 = H} = {g ∈ G: gH = Hg}. Clearly CG(H) ⊆ NG(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, CG(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., Φa: G → G defined by Φa(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 → Φg, where Φg is the inner automorphism induced or given by g, that is, Φg(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 → Φx where Φx(h) = xhx-1 ∀h ∈ H.

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

Φx°Φy ⇒ [° represents composition] ∀h ∈ H: Φx°Φy(h) = Φxy(h)) =[By definition of inner automorphism] Φx(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 = Φxy(h)

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

Recall: CG(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 CG(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

This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. This post relies heavily on the following resources, specially on NPTEL-NOC IITM, Introduction to Galois Theory, Michael Penn, and Contemporary Abstract Algebra, Joseph, A. Gallian.
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.
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