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Hello again,

I just have a few more problems I hope you could help me with,

Ques. 14.14 a) Prove that if a and b are elements of an Abelian group G, with o(a)=M and o(b)=n, then (ab)^mn = e.

Ques. 15.23 Prove that if a, b in Z, then {a,b}={d}, where d is the greatest common divisor of a and b. Formulate a generalization involving {a1,a2,...,an} for a1,a2,...,an in Z.

This question resembles the question in the mini-quiz. Do I show that {d} is a subset of {a,b} and that {a,b} is a subset of {d}. If there is a better way, could you please let me know.

Thanks a lot

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I have another question, I hope you're not too busy.

Ques. 16.12 Verify that if H is a subgroup of an Abelian group G, and a belongs in G, then the right coset of H to which a belongs is the same as the left coset of H to which a belongs.

Thanks again

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14.4. (ab)^{mn}=(ab)(ab)...(ab) mn many times. Because of commutativity, we have (ab)(ab)...(ab)=aa...abb...b with mn many a-s and b-s. So,(ab)^{mn}=a^{mn}b^{mn}=(a^{m})^{n} (b^{n})^{m}=e.e=e.

15.23. I am assuming you meant <{a,b}> = <d>. Your approach is OK. Here is a brief outline. Show <d> is a subset of <{a,b}>: suffices to show that d is in <{a,b}>. But we have a theorem which states that: the GCD of a and b can be written as na+mb for some integers n and m; and that is exactly what we need to show d is in <{a,b}>. Then you show that <{a,b}> is a subset of <d> - easier.

16.12 The right coset is Ha (you multiply everything in H by a to the right); but becasue of commutativity, that is obviously equal to aH where you multiply everything in H by a to the left.