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## Math 401 - 01 Homework (Fall 2018)


Problem Set 05 (complete) Due: 09/24/2018. Board presentation 09/??/2018

1. Chapter 5, problems 6, 8. For all of them find the order and the parity.
2. Chapter 5, problem 10. What is the largest order of an element of $S_8$. Explain.
3. Chapter 5, problems 23, 24.
4. Chapter 5, problem 48.
5. Chapter 5, problem 50. Is $D_5$ a subgroup of $A_5$? Explain.

Problem Set 04 (complete) Due: 09/17/2018. Board presentation: 09/24/2018

1. Chapter 4, problem 74.
2. Chapter 5, problem 2.a.
3. Chapter 5, problem 4.
4. Consider $\alpha\in S_8$ given in disjoint cycle form by $\alpha=(1\ 4\ 5)(3\ 7)$. Write $\alpha$ in array form.

Problem Set 03 (complete) Due: 09/12/2018. Board presentation: 09/17/2018

1. Let $G=\pbr{a}$ be an infinite cyclic group, and $k_1,k_2\in\Z$. Prove that $$\pbr{a^{k_1}}\leq\pbr{a^{k_2}} \textrm{ iff } k_2\mid k_1.$$
2. Let $G=\pbr{a}$ be a cyclic group of order $60$.
1. How many subgroups does $G$ have?
2. Which of them are cyclic?
3. List a generator for each of the cyclic subgroups of $G$.
4. Draw the subgroup lattice of $G$.
3. Prove that a finite group of prime order must be cyclic.
4. Chap. 4, problem 38, 62.
5. Chap. 4, problem 50.