please send handwritten solution for part e Problem 4. Read the following mathematical text and answer the associated comprchension1/3questions at the end.The goal of this text is to prove the following theorem:Theorem 1. Let G be a finite commutative group whose order is a product ofthe form pi x x Pk: with p1,. Pk distinct prime numbers. Then G is a cyclicgroup.Before giving the proof of this theorem, we introduce the following intermediateresults that will be needed:Proposition 2. Let G be a finite commutative group, and let a, beG be twoelements of order m, n respectively. If m and n are coprime, then the product abhas order m x n.Proof of Proposition 2. Let e denote the identity element of G. Since a and bhave order m and n respectively, we have(ab)mn = am"nn = (a")"(b")" = e"e" = e.Let d denote the order of ab. Since (ab)m =e, it follows that d 2. We write the order of G as G| = P xx P, for some distinctprime numbers P Pk. Since the prime numbers p..., Pa are divisors of |G],we can choose elements a,.. a of order p... P respectively. Since G is acommutative group, and the orders p....pa are distinct prime numbers, andin particular pairwise coprime, it follows from Proposition 3 that the producta.a, has order p x.x P. - |G|. Sinceelement of order |G|, it follows that G is cyclic.is a finite group that contains anComprehension questions:(a) Explain how Theorem 1 can be used to show that the multiplicative groupZ, is cyclic.(b) Show that Theorem 1 no longer holds without the condition on the orderof the group G. That is, give an example of a finite commutative group Gwhose order is not a product of distinct primes and such that G is not cyclic.(c) Show that Proposition 2 no longer holds without the condition that G iscommutative. That is, give an example of a finite non-commutative groupG and two elements a, b with order m, n respectively, such that m and n arecoprime but the product ab does not have order mn.(d) The proof of Proposition 2 is currently incomplete, as it never explicitly usesthe hypothesis that G is commutative. Explain what part of the proof usesimplicitly the commutativity of G. Give an example to show how that partof the proof would break down without this hypothesis on G. (e) In the proof of Proposition 2 (last line of page 2), the following statement isnot properly justified:“ It thus follows that the order m of a divides the order d of ab. "Give a complete justification of this statement, by explaining carefully howthis statement is a consequence of what comes before it.(f) A student proposed the following argument to generalise the proof of Theo-rem 1 to the case where G is a commutative group whose order is no longerassumed to be a product of distinct primes:Student's proposed generalisation: Let G be a finite commutative group.Then G is cyclic.Student's proposed proof. The result is true when G| = 1, so we considerthe case where |G| 2. We write a prime decomposition for the order ofG as |G| = pi' x ...x p, for some distinct prime numbers p1, ..., Pk and, p are divisors of G|, we canchoose elements a1,.., ak of order p', ..., p respectively. Since p1,.. , Pkare distinct prime numbers, it follows that pi",...,P are pairwise coprime.Since G is commutative group, and the orders of a1,..., ak are pairwisecoprime, it follows from Proposition 3 that the product a1ak has orderx p = |G]. Since G is a finite group that contains an element of%3Dpositive integers a1,...,ak. Since pi',..........order |G|, it follows that G is cyclic.We know from part (b) that Theorem 1 does not hold without the hypothesison the order of G. Where exactly is the error in the student's proposedproof? Give an example to illustrate how that part of the argument breaksdown without additional assumptions on G, and explain why the problemyou identified in the student's proposed proof is not an issue in the proof ofTheorem 1.

Question

please send handwritten solution for part e Problem 4. Read the following mathematical text and answer the associated comprchension1/3questions at the end.The goal of this text is to prove the following theorem:Theorem 1. Let G be a finite commutative group whose order is a product ofthe form pi x x Pk: with p1,. Pk distinct prime numbers. Then G is a cyclicgroup.Before giving the proof of this theorem, we introduce the following intermediateresults that will be needed:Proposition 2. Let G be a finite commutative group, and let a, beG be twoelements of order m, n respectively. If m and n are coprime, then the product abhas order m x n.Proof of Proposition 2. Let e denote the identity element of G. Since a and bhave order m and n respectively, we have(ab)mn = am"nn = (a")"(b")" = e"e" = e.Let d denote the order of ab. Since (ab)m =e, it follows that d < mn.Let us show that m divides d. We have (ab)"= a"b" a", since b has order n.Thus, the cyclic subgroup (ab) contains a". Moreover, it also contains a" = esince a has order m. Since m and n are coprime by hypothesis, Bézout's Identitygives us two integers k, l e Z such that km+ En = 1. Since the cyclic subgroup(ab) is stable under products and inverses, and contains a" and a", it also contains(a")*(a")' = a*m+en = a'It thus follows that the order m of a divides the order d of ab.=a.Since a and b play symmetric roles, a similar proof shows that n divides the orderd of ab. Thus, both integers n and m divide d. Since n and m are coprime, itfollows that their product mn also divides d. Since we proved earlier that d< mn,it follows that d-mn, which concludes the proof of Proposition 2.One can prove by induction the following generalisation of Proposition 2, theproof of which we will omit:Proposition 3. Let G be a commutative group, and let a,,...,a, € G be k> 2elements of order mcoprime, then the product (a,a) has order m¡ × .…x m.m, respectively. If the integers m m are pairwiseWe are now ready to prove the main theorem:Proof of Theorem 1. The result is true when |G| = 1, so we consider the casewhere |G| > 2. We write the order of G as G| = P xx P, for some distinctprime numbers P Pk. Since the prime numbers p..., Pa are divisors of |G],we can choose elements a,.. a of order p... P respectively. Since G is acommutative group, and the orders p....pa are distinct prime numbers, andin particular pairwise coprime, it follows from Proposition 3 that the producta.a, has order p x.x P. - |G|. Sinceelement of order |G|, it follows that G is cyclic.is a finite group that contains anComprehension questions:(a) Explain how Theorem 1 can be used to show that the multiplicative groupZ, is cyclic.(b) Show that Theorem 1 no longer holds without the condition on the orderof the group G. That is, give an example of a finite commutative group Gwhose order is not a product of distinct primes and such that G is not cyclic.(c) Show that Proposition 2 no longer holds without the condition that G iscommutative. That is, give an example of a finite non-commutative groupG and two elements a, b with order m, n respectively, such that m and n arecoprime but the product ab does not have order mn.(d) The proof of Proposition 2 is currently incomplete, as it never explicitly usesthe hypothesis that G is commutative. Explain what part of the proof usesimplicitly the commutativity of G. Give an example to show how that partof the proof would break down without this hypothesis on G. (e) In the proof of Proposition 2 (last line of page 2), the following statement isnot properly justified:“ It thus follows that the order m of a divides the order d of ab. "Give a complete justification of this statement, by explaining carefully howthis statement is a consequence of what comes before it.(f) A student proposed the following argument to generalise the proof of Theo-rem 1 to the case where G is a commutative group whose order is no longerassumed to be a product of distinct primes:Student's proposed generalisation: Let G be a finite commutative group.Then G is cyclic.Student's proposed proof. The result is true when G| = 1, so we considerthe case where |G| > 2. We write a prime decomposition for the order ofG as |G| = pi' x ...x p, for some distinct prime numbers p1, ..., Pk and, p are divisors of G|, we canchoose elements a1,.., ak of order p', ..., p respectively. Since p1,.. , Pkare distinct prime numbers, it follows that pi",...,P are pairwise coprime.Since G is commutative group, and the orders of a1,..., ak are pairwisecoprime, it follows from Proposition 3 that the product a1ak has orderx p = |G]. Since G is a finite group that contains an element of%3Dpositive integers a1,...,ak. Since pi',..........order |G|, it follows that G is cyclic.We know from part (b) that Theorem 1 does not hold without the hypothesison the order of G. Where exactly is the error in the student's proposedproof? Give an example to illustrate how that part of the argument breaksdown without additional assumptions on G, and explain why the problemyou identified in the student's proposed proof is not an issue in the proof ofTheorem 1.

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