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## Math 401 - 01 Homework (Fall 2018)


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.