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Exercise 15.5.33. (a) Find all cyclic subgroups of the symmetry group of the square (i.e. D₁) by finding (a) for every element a € D₁. (b) Find all nontrivial proper subgroups of D₁ (You may follow the proce- dure used in Example 15.5.32 if you wish.) (c) Show that at least one of the subgroups in (b) is abelian and not cyclic.

Exercise 15.5.33. (a) Find all cyclic subgroups of the symmetry group of the square (i.e. D₁) by finding (a) for every element a € D₁. (b) Find all nontrivial proper subgroups of D₁ (You may follow the proce- dure used in Example 15.5.32 if you wish.) (c) Show that at least one of the subgroups in (b) is abelian and not cyclic.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Exercise 15.5.33. (a) Find all cyclic subgroups of the symmetry group of the square (i.e. D₁) by finding (a) for every element a € D₁. (b) Find all nontrivial proper subgroups of D₁ (You may follow the proce- dure used in Example 15.5.32 if you wish.) (c) Show that at least one of the subgroups in (b) is abelian and not cyclic.
Transcribed Image Text:Exercise 15.5.33. (a) Find all cyclic subgroups of the symmetry group of the square (i.e. D₁) by finding (a) for every element a € D₁. (b) Find all nontrivial proper subgroups of D₁ (You may follow the proce- dure used in Example 15.5.32 if you wish.) (c) Show that at least one of the subgroups in (b) is abelian and not cyclic.
Example 15.5.32. We showed in Example 15.5.29 that D3 has 4 cyclic subgroups, and that every element of D3 is in at least one of these subgroups. Proposition 15.5.30 shows that, for example, any subgroup containing p₁ must also contain id and p2, since (p₁) = {id, p1, p2}. Let's try to find a larger subgroup H C D3 that contains p₁. If we add any other element (which must be µ for some k = 1,2 or 3), then we must also add pik and P2k, which means that H contains all 6 elements of D3. It follows that H = D3. Similarly, if we try to find a subgroup K that contains by adding another reflection µ(j ‡k), we find that i and jk must also be in K, which means that p₁ must also be in K. But we've just finished shown that if p₁ € K and μ € K, then K = G. It follows that the only 538 CHAPTER 15 INTRODUCTION TO GROUPS Oo proper nontrivial subgroups of D3 are the four cyclic subgroups shown in Figure 15.5.1.
Transcribed Image Text:Example 15.5.32. We showed in Example 15.5.29 that D3 has 4 cyclic subgroups, and that every element of D3 is in at least one of these subgroups. Proposition 15.5.30 shows that, for example, any subgroup containing p₁ must also contain id and p2, since (p₁) = {id, p1, p2}. Let's try to find a larger subgroup H C D3 that contains p₁. If we add any other element (which must be µ for some k = 1,2 or 3), then we must also add pik and P2k, which means that H contains all 6 elements of D3. It follows that H = D3. Similarly, if we try to find a subgroup K that contains by adding another reflection µ(j ‡k), we find that i and jk must also be in K, which means that p₁ must also be in K. But we've just finished shown that if p₁ € K and μ € K, then K = G. It follows that the only 538 CHAPTER 15 INTRODUCTION TO GROUPS Oo proper nontrivial subgroups of D3 are the four cyclic subgroups shown in Figure 15.5.1.
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