~~META:title=Previous Homework~~

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===== Math 401 - 01 Previous Homework (Fall 2018)=====
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**Problem Set 10** (complete) Due: 11/06/2018. Board presentation: 11/20/2018
  - Let $G$ be a group, and $H,K\leq G$.  
    - Prove that if $HK=KH$, then $HK\leq G$. 
    - Prove that if $H\leq N_G(K)$, then $HK\leq G$.
  - 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: \[ |C|=[G:N_G(H)]. \]
  - 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?
  - 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
  - Prove that, up to isomorphism, the direct product operation is commutative and associative.
  - 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 $H\oplus K$.
  - Let $G$ be a group, and $H,N\leq G$. Prove that:
    - If $N\normaleq G$, then $HN\leq G$.
    - If both $H,N\normaleq G$, then $HN\normaleq G$.
  - 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
  - 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)
  - 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)
  - Combine the previous problem with problem 3 in Problem Set 6.
  - 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
  - Prove Thm. 6.2.3, Thm. 6.3.2, Thm. 10.2.3.  Combine all three proofs into one.
  - Chapter 10, problems 8, 10.

**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 $60$. 
    - How many subgroups does $G$ have? 
    - Which of them are cyclic? 
    - List a generator for each of the cyclic subgroups of $G$.
    - 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, 

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