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In Exercises 21 through 25, proceed as in Example 9.13 to find all abelian groups, up to isomorphism, of the given order. For each group, find the invariant factors and find an isomorphic group of the form indicated in Theorem 9.14. 21. Order 8 22. Order 16 23. Order 32 24/ Order 720 25. Order 1089

In Exercises 21 through 25, proceed as in Example 9.13 to find all abelian groups, up to isomorphism, of the given order. For each group, find the invariant factors and find an isomorphic group of the form indicated in Theorem 9.14. 21. Order 8 22. Order 16 23. Order 32 24/ Order 720 25. Order 1089

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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#24 only just the invariant factors by use of table

Here is the transcribed text and explanation of the diagrams from the image: --- **22. Order 8** **23. Order 32** **24. Order 720** **25. Order 1089** b. Let \( p_1 \) and \( p_2 \) be distinct prime numbers. Use the table you created to find the number of abelian groups, up to isomorphism, of the given order. i. \( p_1p_2 \) ii. \( p_1^2 \) iii. \( p_1^3p_2^4 \) **26.** How many abelian groups (up to isomorphism) are there of order \( 24 \)? (5) **27.** Following the idea suggested in Exercise 26, let \( r \) and \( s \) be relatively prime positive integers. Show that if there are \( r \) (up to isomorphism) abelian groups of order \( r \), and \( s \) (up to isomorphism) abelian groups of order \( s \), then there are \( rs \) (up to isomorphism) abelian groups of order \( rs \). **28.** Use Exercise 27 to determine the number of abelian groups of order \( 720 \). **29.** If \( n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k} \) where \( p_i \) are distinct prime numbers, let *number of groups* = list the number of abelian groups of the form indicated in the first row of the table. | \# | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |----|---|---|---|---|---|---|---| | number of groups | | | | | | | | 16. Are the groups \( \mathbb{Z}_6 \times \mathbb{Z}_{10} \) and \( \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_5 \) isomorphic? Why or why not? 17. Are the groups \( \mathbb{Z}_{18} \times \mathbb{Z}_{12} \) and \( \mathbb{
Transcribed Image Text:Here is the transcribed text and explanation of the diagrams from the image: --- **22. Order 8** **23. Order 32** **24. Order 720** **25. Order 1089** b. Let \( p_1 \) and \( p_2 \) be distinct prime numbers. Use the table you created to find the number of abelian groups, up to isomorphism, of the given order. i. \( p_1p_2 \) ii. \( p_1^2 \) iii. \( p_1^3p_2^4 \) **26.** How many abelian groups (up to isomorphism) are there of order \( 24 \)? (5) **27.** Following the idea suggested in Exercise 26, let \( r \) and \( s \) be relatively prime positive integers. Show that if there are \( r \) (up to isomorphism) abelian groups of order \( r \), and \( s \) (up to isomorphism) abelian groups of order \( s \), then there are \( rs \) (up to isomorphism) abelian groups of order \( rs \). **28.** Use Exercise 27 to determine the number of abelian groups of order \( 720 \). **29.** If \( n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k} \) where \( p_i \) are distinct prime numbers, let *number of groups* = list the number of abelian groups of the form indicated in the first row of the table. | \# | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |----|---|---|---|---|---|---|---| | number of groups | | | | | | | | 16. Are the groups \( \mathbb{Z}_6 \times \mathbb{Z}_{10} \) and \( \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_5 \) isomorphic? Why or why not? 17. Are the groups \( \mathbb{Z}_{18} \times \mathbb{Z}_{12} \) and \( \mathbb{
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