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(a) Following the same approach as in the proof of Proposition 2, show that for an integer m> 1, the additive group Zm either contains no element of order 3 or contains exactly two elements of order 3.

(a) Following the same approach as in the proof of Proposition 2, show that for an integer m> 1, the additive group Zm either contains no element of order 3 or contains exactly two elements of order 3.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please send solution for part a b c only handwritten solution accepted
We are now ready to prove Theorem 1: Proof of Theorem 1. We know that an element k E Zyn belongs to Z, if and only if k and 2" are coprime, that is, if and only ifk is odd. In particular, the elements a := 2"-1 -1 and b:= 2n-1 +1 belong to Z, and we have a² = (2"-I - T)² : 2n x2n-2 + T = T, %3D 2n and (2"-1+1)2: 2" x2n-2 + ī =T. 2n Thus, Z contains at least two distinct elements of order 2. Since cyclic groups contain at most one element of order 2 by Proposition 2, this implies that Z is not cyclic. Comprehension questions: (a) Following the same approach as in the proof of Proposition 2, show that for an integer m > 1, the additive group Zm either contains no element of order 3 or contains exactly two elements of order 3. (b) Explain how Proposition 3 can be used to show that the multiplicative group Z is not cyclic. (c) In the proof of Proposition 3, the following statement is not justified: "However, since a is non-trivial, f(x) cannot have order 1." Properly justify this statement. (d) Show that Theorem 1 does not hold for = 1 and n 2. That is, show that the multiplicative groups Z and Z are cyclic. (e) The proof of Theorem 1 is thus currently incomplete, as it does not explicitly use the hypothesis that n > 3. In what parts of the proof is that assumption implicitly used?
Transcribed Image Text:We are now ready to prove Theorem 1: Proof of Theorem 1. We know that an element k E Zyn belongs to Z, if and only if k and 2" are coprime, that is, if and only ifk is odd. In particular, the elements a := 2"-1 -1 and b:= 2n-1 +1 belong to Z, and we have a² = (2"-I - T)² : 2n x2n-2 + T = T, %3D 2n and (2"-1+1)2: 2" x2n-2 + ī =T. 2n Thus, Z contains at least two distinct elements of order 2. Since cyclic groups contain at most one element of order 2 by Proposition 2, this implies that Z is not cyclic. Comprehension questions: (a) Following the same approach as in the proof of Proposition 2, show that for an integer m > 1, the additive group Zm either contains no element of order 3 or contains exactly two elements of order 3. (b) Explain how Proposition 3 can be used to show that the multiplicative group Z is not cyclic. (c) In the proof of Proposition 3, the following statement is not justified: "However, since a is non-trivial, f(x) cannot have order 1." Properly justify this statement. (d) Show that Theorem 1 does not hold for = 1 and n 2. That is, show that the multiplicative groups Z and Z are cyclic. (e) The proof of Theorem 1 is thus currently incomplete, as it does not explicitly use the hypothesis that n > 3. In what parts of the proof is that assumption implicitly used?
We are now ready to prove Theorem 1: Proof of Theorem 1. We know that an element kEZn belongs to Z if and only if k and 2" are coprime, that is, if and only ifk is odd. In particular, the elements a := 2"-1 -1 and b:= 2n-1 +1 belong to Z, and we have a² = (2"-I - T)² : + T = T, 2n x2n-2 2n %3D %3D and (2"-1+1)2: + T = T. 2n x2n-: 2n Thus, Z contains at least two distinct elements of order 2. Since cyclic groups contain at most one element of order 2 by Proposition 2, this implies that Z is not cyclic. Comprehension questions: (a) Following the same approach as in the proof of Proposition 2, show that for an integer m > 1, the additive group Zm either contains no element of order 3 or contains exactly two elements of order 3. (b) Explain how Proposition 3 can be used to show that the multiplicative group Z is not cyclic. (c) In the proof of Proposition 3, the following statement is not justified: "However, since x is non-trivial, f(x) cannot have order 1." Properly justify this statement. (d) Show that Theorem 1 does not hold for = 1 and n = 2. That is, show that the multiplicative groups Z and Z are cyclic. (e) The proof of Theorem 1 is thus currently incomplete, as it does not explicitly use the hypothesis that n > 3. In what parts of the proof is that assumption implicitly used?
Transcribed Image Text:We are now ready to prove Theorem 1: Proof of Theorem 1. We know that an element kEZn belongs to Z if and only if k and 2" are coprime, that is, if and only ifk is odd. In particular, the elements a := 2"-1 -1 and b:= 2n-1 +1 belong to Z, and we have a² = (2"-I - T)² : + T = T, 2n x2n-2 2n %3D %3D and (2"-1+1)2: + T = T. 2n x2n-: 2n Thus, Z contains at least two distinct elements of order 2. Since cyclic groups contain at most one element of order 2 by Proposition 2, this implies that Z is not cyclic. Comprehension questions: (a) Following the same approach as in the proof of Proposition 2, show that for an integer m > 1, the additive group Zm either contains no element of order 3 or contains exactly two elements of order 3. (b) Explain how Proposition 3 can be used to show that the multiplicative group Z is not cyclic. (c) In the proof of Proposition 3, the following statement is not justified: "However, since x is non-trivial, f(x) cannot have order 1." Properly justify this statement. (d) Show that Theorem 1 does not hold for = 1 and n = 2. That is, show that the multiplicative groups Z and Z are cyclic. (e) The proof of Theorem 1 is thus currently incomplete, as it does not explicitly use the hypothesis that n > 3. In what parts of the proof is that assumption implicitly used?
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