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 \(

Answered: Image /qna-images/answer/c69e907f-6d75-4957-97d7-3dc040fbd429.jpg

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.

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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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 $; and
2. 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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