Previous Homework [Binghamton University Department of Mathematical Sciences]

Math 401 - 01 Previous Homework (Fall 2018)

$\newcommand{\aut}{\textrm{Aut}} \newcommand{\inn}{\textrm{Inn}} \newcommand{\sub}{\textrm{Sub}} \newcommand{\cl}{\textrm{cl}} \newcommand{\join}{\vee} \newcommand{\bigjoin}{\bigvee} \newcommand{\meet}{\wedge} \newcommand{\bigmeet}{\bigwedge} \newcommand{\normaleq}{\unlhd} \newcommand{\normal}{\lhd} \newcommand{\union}{\cup} \newcommand{\intersection}{\cap} \newcommand{\bigunion}{\bigcup} \newcommand{\bigintersection}{\bigcap} \newcommand{\sq}[2][\ ]{\sqrt[#1]{#2\,}} \newcommand{\pbr}[1]{\langle #1\rangle} \newcommand{\ds}{\displaystyle} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\A}{\mathbb{A}} \newcommand{\F}{\mathbb{F}} \newcommand{\T}{\mathbb{T}} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\imp}{\Rightarrow} \newcommand{\rimp}{\Leftarrow} \newcommand{\pinfty}{1/p^\infty} \newcommand{\power}{\mathcal{P}} \newcommand{\calL}{\mathcal{L}} \newcommand{\calC}{\mathcal{C}} \newcommand{\calN}{\mathcal{N}} \newcommand{\calB}{\mathcal{B}} \newcommand{\calF}{\mathcal{F}} \newcommand{\calR}{\mathcal{R}} \newcommand{\calS}{\mathcal{S}} \newcommand{\calU}{\mathcal{U}} \newcommand{\calT}{\mathcal{T}} \newcommand{\gal}{\textrm{Gal}} \newcommand{\isom}{\approx} $

Problem Set 06 (complete) Due: 10/08/2018. Board presentation 10/10/2018

Chapter 7, problem 8.
Chapter 7, problem 22.
Let $G$ be a finite group, and $p$ the smallest prime divisor of $|G|$. If $p^2\not\mid|G|$, then $G$ has at most one subgroup of index $p$. (Hint:Look at Example 6 on page 144)
Chapter 7, problem 12. Generalize.
Chapter 7, problem 48.

Problem Set 05 (complete) Due: 09/24/2018. Board presentation 09/28/2018

Chapter 5, problems 6, 8. For all of them find the order and the parity.
Chapter 5, problem 10. What is the largest order of an element of $S_8$. Explain.
Chapter 5, problems 23, 24.
Chapter 5, problem 48.
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

Chapter 4, problem 74.
Chapter 5, problem 2.a.
Chapter 5, problem 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

Let $G=\pbr{a}$ be an infinite cyclic group, and $k_1,k_2\in\Z$. Prove that $$\pbr{a^{k_1}}\leq\pbr{a^{k_2}} \textrm{ iff } k_2\mid k_1.$$
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 $G$.
Prove that a finite group of prime order must be cyclic.
Chap. 4, problem 38, 62.
Chap. 4, problem 50.

Problem Set 02 (complete) Due: 09/04/2018. Board presentation: 09/12/2018

Chap. 3, problems 4, 13, 20, 64
Chap. 3, problems 6, 50
Let $G$ be a group in which every non-identity element has order 2. Prove that $G$ must be Abelian.
Chap. 3, problem 34. what can you say about the union of two subgroups?

Problem Set 01 (complete) Due: 08/27/2018. Board presentation: 08/31/2018

Page 38, prob. 18. What happens if you replace each H with an A?
Page 39, prob. 22. Explain.
Page 54, prob. 4.
Page 56, prob. 22. Compare with problem 13 on page 38.

Additional problems to look at: 1.13 p.38, 1.14 p.38, 2.5 p.54,

Home

User Tools

Site Tools

Sidebar

Math 401 - 01 Previous Homework (Fall 2018)

Page Tools