Could you explain how to show 5.30 in detail? I also included lists of definitions and theorems in the book as a reference. **5.22.** Find the invariant factors for each of the following groups.1. \( \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_9 \times \mathbb{Z}_{25} \times \mathbb{Z}_{11} \times \mathbb{Z}_{121} \)2. \( \mathbb{Z}_4 \times \mathbb{Z}_8 \times \mathbb{Z}_8 \times \mathbb{Z}_{16} \times \mathbb{Z}_5 \times \mathbb{Z}_{25} \times \mathbb{Z}_{49} \)**5.30.** Let \( G \) be a finite abelian group. Suppose that, for every \( n \in \mathbb{N} \), there are at most \( n \) elements \( a \in G \) satisfying \( a^n = e \). Show that \( G \) is cyclic. **Definition 5.4.** Let \( G \) be a nontrivial finite abelian group, and say that \( G = H_1 \times H_2 \times \cdots \times H_k \), where each \( H_i \) is cyclic of order \( p_i^{n_i} \), for some prime \( p_i \) and positive integer \( n_i \). Then the elementary divisors of \( G \) are the numbers \( p_1^{n_1}, p_2^{n_2}, \ldots, p_k^{n_k} \), where the order in this list is irrelevant, but each number must be listed as many times as it occurs. The trivial group has no elementary divisors.**Definition 5.5.** Let \( G \) be an abelian group and \( n \) a positive integer. Then we write \( G^n = \{a^n : a \in G\} \).**Lemma 5.7.** Let \( G \) and \( H \) be abelian groups and \( n \) a positive integer. Then:1. \( G^n \) is a subgroup of \( G \); and2. if \( \alpha : G \to H \) is an onto homomorphism, then \( \alpha(G^n) = H^n \).**Theorem 5.6.** Let \( G \) and \( H \) be finite abelian groups. Then \( G \) and \( H \) are isomorphic if and only if they have the same elementary divisors.**Theorem 5.7 (Invariant Factor Decomposition).** Suppose that \( G \) is a nontrivial finite abelian group. Then \( G = H_1 \times H_2 \times \cdots \times H_k \), where each \( H_i \) is a cyclic subgroup of \( G \) of order \( m_i \), with \( m_1 > 1 \) and \( m_i|m_{i+1} \), for \( 1 \leq i < k \).**Definition 5.6.** If \( G \) is isomorphic to \( \mathbb{Z}_{m_1} \times \cdots \times \mathbb{Z}_{m_k} \), where \(

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Answered: 5.30. Let G be a finite abelian group.…

5.30. Let G be a finite abelian group. Suppose that, for every n e N, there are at most n elements a e G satisfying a" = e. Show that G is cyclic.

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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