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Exercise 8.2. (a) Suppose G₁ = (gi) for i = 1, 2, ..., m, |G₁| <∞. Prove that ged(|G₁|, |G₁|) = 1 for i ‡ j if and only if m m i=1 i=1 G₁ = ((91, 92, ..., 9m)). (b) Suppose that G; is cyclic. Prove that each G₁ is cyclic. (c) Prove that ZZ is not cyclic. This shows that the converse of part (b) is false.

Exercise 8.2. (a) Suppose G₁ = (gi) for i = 1, 2, ..., m, |G₁| <∞. Prove that ged(|G₁|, |G₁|) = 1 for i ‡ j if and only if m m i=1 i=1 G₁ = ((91, 92, ..., 9m)). (b) Suppose that G; is cyclic. Prove that each G₁ is cyclic. (c) Prove that ZZ is not cyclic. This shows that the converse of part (b) is false.

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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Exercise 8.2. (a) Suppose G₁ = (gi) for i = 1, 2, ..., m, |G₁| <∞. Prove that ged(|G₁|, |G₁|) = 1 for i ‡ j if and only if m m i=1 i=1 G₁ = Gi (b) Suppose that G; is cyclic. Prove that each G₂ is cyclic. i ((91, 92, ..., 9m)). (c) Prove that ZZ is not cyclic. This shows that the converse of part (b) is false.
Transcribed Image Text:Exercise 8.2. (a) Suppose G₁ = (gi) for i = 1, 2, ..., m, |G₁| <∞. Prove that ged(|G₁|, |G₁|) = 1 for i ‡ j if and only if m m i=1 i=1 G₁ = Gi (b) Suppose that G; is cyclic. Prove that each G₂ is cyclic. i ((91, 92, ..., 9m)). (c) Prove that ZZ is not cyclic. This shows that the converse of part (b) is false.
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