Beneath this mask there is more than flesh. Beneath this mask there is an idea, Mr. Creedy, and ideas are bulletproof, V for Vendetta
Definition. Let G be a group, and A 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 A. Formally, CG(A) = {g ∈ G | gag-1 = a ∀a ∈ A} = {g ∈ G | ga = ag ∀ a ∈ A}.
Proposition. CG(A) is a subgroup of G.
Proof.
Example. In an Abelian group G, CG(A) = G, for all subsets A.
Definition. The center of a group, G, is the set of elements that commute with all the elements of the group. Z(G) = {g ∈ G | gx = xg, ∀x ∈ G}.
Proposition. Z(G) is a subgroup of G, Z(G) ≤ G.
Proof: Z(G) = CG(G), so the argument previously shown proves Z(G) ≤ G.
As an exercise, let’s prove it directly with the Two-Step Subgroup Test.
gAg-1 = {gag-1 | a ∈ A}. The normalizer of A in the group G is defined as NG(A) = {g ∈ G: gAg-1 = A} = {g ∈ G: gA = Ag}. Clearly CG(A) ⊆ NG(A) and both are subgroups of G.
Examples.
The center of an Abelian group, G, is the group itself. CG(A) = NG(A) = G for any subset A of G.
The center of the quaternion group, Q8 is {e, a2}.
Q8 = ⟨a, b: a4 = e, b2 = a2, aba = b⟩, so in particular Q8 is generated by both a and b.
Let x ∈ Q8, x = aibj for some i ∈ {0, 1, 2, 3} and j ∈ {0, 1}, x ∈ Z(Q8) ↭ xa = ax ∧ xb = bx
xa = ax ↭ aibja = aaibj ↭ aibja = ai+1bj ↭ [By cancellation laws] bja = abj.
For j = 1, ba = ab ⇒ [aba = b ⇒ ab = ba-1] ba = ba-1 ⇒ a = a-1 ⇒ a2 = e⊥ (the order of a is 4, not 2) ⇒ x = ai for some i ∈ {0, 1, 2, 3}.
xb = bx ↭ aib = bai ↭ [Product of Generating Elements of Quaternion Group, Q8 satisfies: bak = a-kb.] aib = a-ib ↭ ai = a-i ↭ a2i = e = a4. Therefore, either i = 0 (x = a0 = e) or 2i = 4, x = ai = a2.
x ∈ ℤ(Dn) iff xα = αx and xβ = βx.
x ∈ ℤ(Dn) = ⟨α, β⟩ ⇒ x = αiβj ⇒ xα = αx ↭ αiβjα = ααiβj = αi+1βj ↔ αiβjα = αi+1βj ↭ [Cancellation laws] βjα = αβj
In particular j = 1, βα = αβ [βαβ = α-1 ⇒ βα = α-1β-1 = αβ] Taking into consideration only the last equality, α-1β-1 = αβ ⇒ [β2 = e ⇒ β = β-1] α-1β = αβ ⇒ [Cancellation laws] α-1 = α ⇒ α2 = e ⊥. Therefore, x cannot be a reflection (all reflections are generated by αiβ, 0 ≤ i ≤ n-1, j = 1), it necessary needs to be a rotation, x = αi.
x ∈ ℤ(Dn) ⇒ xβ = βx ⇒ αiβ = βαi ⇒ [Property: αkβ = βαn-k ↭ αn-kβ = βαk] αiβ = αn-iβ = [αn = e] α-iβ ⇒ αiβ = α-iβ ⇒ [Cancellation laws] αi = α-i ⇒ α2i = e ⇒ i = 0 (x = α0 = e) or i = n/2 and n is even.
Therefore, ℤ(Dn) = $ \begin{cases} e, n~is~ odd\\\\ e~ and~ α^{\frac{n}{2}}, n~is~ even \end{cases}$
Let’s prove that ND8(A) = D8. First, CD8(A) = A ⊆ ND8(A).
sAs-1 = {s1s-1, srs-1, sr2s-1, sr3s-1} =🔑 {1, r3, r2, r} = A ⇒ s ∈ ND8(A).
🔑 srs-1 = [sr = r-1s] r-1ss-1 = r-1 = [r4=1] r3. sr2s-1 = sr2ss-2 [rks = sr4-k] ssr2s-2 = [s2 = s-2 = e] r2. sr3s-1 = sr3ss-2 = s2rs-2 = r.
r, s ∈ ND8(A) ≤ D8 ⇒ [Every subgroup is closed under the group operation] sirj ∈ ND8(A) ⇒ ND8(A) = D8.