User Tools

Site Tools


Math 401 - 01 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 11 (complete) Due: 11/20/2018. Board presentation: 11/??/2018

  1. Chapter 12, problem 18. Moreover, if $R$ is commutative, then $S$ is an ideal of $R$.
  2. Chapter 12, problem 28.
  3. Chapter 13, problem 52.
  4. Chapter 13, problem 34.

Problem Set 10 (complete) Due: 11/06/2018. Board presentation: 11/20/2018

  1. Let $G$ be a group, and $H,K\leq G$.
    1. Prove that if $HK=KH$, then $HK\leq G$.
    2. Prove that if $H\leq N_G(K)$, then $HK\leq G$.
  2. Let $G$ be a group, $H\leq G$, and $C=\{gHg^{-1}|g\in G\}$ the set of all conjugates of $H$ in $G$. Prove that: \[ |C|=[G:N_G(H)]. \]
  3. Let $G$ be a group of order $120$. What are the possible values of $n_2$, $n_3$, and $n_5$, i.e. the number of Sylow 2-subgroups, the number of Sylow 3-subgroups and the number of Sylow 5-subgroups?
  4. How many groups of order $6727$ are there? Describe them. Justify your answers. Show all your work.

Problem Set 09 (complete) Due: 10/29/2018. Board presentation: 11/02/2018

  1. Prove that, up to isomorphism, the direct product operation is commutative and associative.
  2. Give an example of a group $G$ with two subgroups $H$ and $K$ such that $HK=G$, $H\intersection K=1$, $K\normaleq G$, but $G$ is not isomorphic to the direct product $H\oplus K$.
  3. Let $G$ be a group, and $H,N\leq G$. Prove that:
    1. If $N\normaleq G$, then $HN\leq G$.
    2. If both $H,N\normaleq G$, then $HN\normaleq G$.
  4. Make a list of all abelian groups of order $2736$. Express each of them using the “elementary divisors” form and the “invariant factors” form.

Problem Set 08 (complete) Due: 10/22/2018. Problem 4 may be resubmitted by 10/24/2018. Board presentation 10/29/2018

  1. Let $n\in\N$ and $H\leq G$ such that $H$ is the only subgroup of $G$ of order $n$. Show that $H\normaleq G$. (Do not assume that $G$ is finite)
  2. Let $n\in\N$ and $H\leq G$ such that $H$ is the only subgroup of $G$ of index $n$. Show that $H\normaleq G$. (Do not assume that $G$ is finite)
  3. Combine the previous problem with problem 3 in Problem Set 6.
  4. Let $p,q$ be primes such that $p < q$ and $p\not\mid (q-1)$. Prove that, up to isomorphism, there is only one group of order $pq$. (Hint: Use example 17, page 203, as a guide. No use this example, you may use the extra assumption that $(p-1)\not\mid (q-1)$, or equivalently that $(p-1)\not\mid (pq-1)$.)

Problem Set 07 (complete) Due: 10/15/2018. Board presentation 10/29/2018

  1. Prove Thm. 6.2.3, Thm. 6.3.2, Thm. 10.2.3. Combine all three proofs into one.
  2. Chapter 10, problems 8, 10.

Previous Homework


people/fer/401ws/fall2018/homework.txt · Last modified: 2018/11/19 10:59 by fer