(2) In the group G =Z, let H =(4) and K =(8) %3D (i) (ii) List the elements inG /H, GIK and in H/K , showing the elements in each coset. List the cosets in (G /K)/(H IK), showing the elements in each coset. (iii) Give the correspondence between G/H and (G / K)/(H /K)using the Third Isomorphism Theorem.

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Q: 8. Let (G, *) be a group. Define the center of G by Z(G) := {x € G: x * a = a * x, Va E G}. The set…

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A: “Since you have posted multiple questions, we will provide the solution only to the first question…

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Q: is a group with identity (eg, eH).

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A: Good question. Have answered all the details in step 2.

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Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D %3D where a = (1 2 3 4)…

A: