Q.2 a). If G is an abelian group that contains a pair of cyclic subgroups of order 2, show that G must contain a subgroup of order .4. Does this subgroup have to be cyclic? b) Find the centre of .D8. What about the centre ofD,0 ? What is the centre of Dn?

Q: HWGrouPA Prove the fallouwing identity. secA.SecB.CSCA.CSCB SectAtB) CSCA.CSCB-SecA. secB

A: sec(A+B)

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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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A: Basic theory regarding normal subgroups from algebra will be used

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A: Find all subgroups of the dihedral group D3 with a detailed explanation.

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Q: Exercises 10 Find the in the alternating group Agi conjagucy desses

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Q: C e 4ch N u ra e eu g dt A a. te t calez let e e sa en es é (: aa'= e) guess : a² = e| (a? = Te -Hhy…

A: Notation used in solution: |a| denotes order of element a not modulus of a. e is identity element .…

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A: Consider the given group G with presentation, x,y:x4=1, x2=y2, xy=yx-1 it can be observed that it is…

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Q: 6.9. Let G = Cn be a cyclic group of order n, and let X = {x₁,...,n} be a set containing n elements.…

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Q: 3. Consider o = (3456) € S, and let H be the subgroup of S, generated by o. Justify all of your…

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Q: (a) Make addition and multiplication tables for Z/3Z.

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

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A: I have used the definition of subgroup generated by a subset.

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Q: elements. Why is Dg not isomorphic to Z/8Z?

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A: Given below solution Explanation:Step 1: Step 2: Step 3: Step 4:

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Q: Show that if aEG, where G is a group and |a| = n then : %3D a' = a' if and only if n divides. -j

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Q: Q3: Prove or disprove any four of the following 1.Any abelain group is simple.

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…