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  3. let g be a group. the center of g is the set z(g)

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

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Advance Math

Let G be a group. The center ofG is the setZ(G) = {g∈G|gh = hg∀h∈G}. Fora∈G, the centralizer ofa is the setC(a)={g∈G |ga=ag }

(a)Prove that Z(G) is anabelian subgroup of G.

(b)Compute the center of D4.

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

â—‹n: no shuffling occurred

â—‹s12: swap the first and second items

â—‹s13: swap the first and third items

â—‹s23: swap the second and third items

â—‹m1: move the last item to the front

â—‹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

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