Let G be a group. The center of G is the set Z(G) = {g∈G |gh...

Let G be a group. The center of G is the set Z(G) = {g∈G |gh = hg ∀h∈G}. For a∈G, the centralizer of a is the set C(a) ={g∈G |ga =ag }

(a)Prove that Z(G) is an abelian subgroup of G.

(b)Compute the center of D₄.

(c)Compute the center of the group G of the shuffles of three objects x₁,x₂,x₃.

○n: no shuffling occurred

○s₁₂: swap the first and second items

○s₁₃: swap the first and third items

○s₂₃: swap the second and third items

○m₁: move the last item to the front

○m₂: move the front item to the end

(d)Compute the center of GL₂(R).

(e)Prove that Z(G) = ∩_a∈GC(a).

please explain every subquestion

Expert Solution

Colby Messinger answered 1 year ago

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Let G be a group. The center of G is the set Z(G) = {g∈G |gh...

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