If H is a subgroup of G, then by the centralizer C(H) of H we mean the set {x in G | xh = hx all h in H} Prove that C(H) is a subgroup of G.
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For h ∈ H we have
eh = h = he
⇒ the identity element e ∈ C(H).
So C(H) is non-empty.
For c ∈ C(H) and h ∈ H we have
ch = hc
⇒ c⁻¹chc⁻¹ = c⁻¹hcc⁻¹
⇒ hc⁻¹ = c⁻¹h
⇒ c⁻¹ ∈ C(H).
So C(H) is closed under inversion.
For c₁,c₂ ∈ C(H) and h ∈ H we have
c₁c₂h = c₁hc₂ = hc₁c₂
⇒ c₁c₂ ∈ C(H).
So C(H) is closed under the product.
→ C(H) is a subgroup of G.
eh = h = he
⇒ the identity element e ∈ C(H).
So C(H) is non-empty.
For c ∈ C(H) and h ∈ H we have
ch = hc
⇒ c⁻¹chc⁻¹ = c⁻¹hcc⁻¹
⇒ hc⁻¹ = c⁻¹h
⇒ c⁻¹ ∈ C(H).
So C(H) is closed under inversion.
For c₁,c₂ ∈ C(H) and h ∈ H we have
c₁c₂h = c₁hc₂ = hc₁c₂
⇒ c₁c₂ ∈ C(H).
So C(H) is closed under the product.
→ C(H) is a subgroup of G.