Department of Mathematics and Statistics, Binghamton University people:fer:401ws:fall2018

Daily topics

2018-11-19T08:57:20-04:00

Math 401 - 01 Daily Topics - part 2 (Fall 2018)

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Week 7	Topics
10/01/2018	Test 1
10/02/2018	Lagrange's corollary 1
	Orbit-Stabilizer theorem
	Examples: cube, truncated icosahedron (soccer ball)
10/03/2018	Corollaries 2-5 to Lagrange's theorem
	Addendum to cor 3: moreover, there is a unique group of order $p$, up to isomorphism.
	Thm. 7.2
	Example 6, p.144
	Corollary: if $p$ is the smallest prime divisor of $\|G\|$ and $p^2$ does not divide $\|G\|$, then $G$ has at most one subgroup of index $p$ (HW)
10/05/2018	Thm. 7.3

Week 8	Topics
10/08/2018	Test 1 returned and reviewed
	Prop: if $\varphi:G\to H$ is an isomorphism, then so is $\varphi^{-1}H\to G$.
	Prop: “isomorphic to” is an equivalence relation
	Thm. 6.1 Cayley's theorem
	$\aut(G)$, $\inn(G)$
10/09/2018	Thm 6.4 $\aut(G)$ is a group and $\inn(G)$ is a subgroup of $\aut(G)$
	Example: $\inn(D_4) \isom K_4$
	Prop: Let $G = <a>$ cyclic and $H$ a group
	1. A homom $\varphi:G\to H$ is uniquely determined by $\varphi(a)$.
	2. If $G$ has order $n$ and $u\in H$ has order $d$ where $d\|n$, then there is (unique) homomorphism $\varphi:G\to H$ s.t. $\varphi(a)=u$. Moreover, $\varphi$ is injective iff $d=n$.
	3. If $G$ has infinite order and $u\in H$, then there is (unique) homomorphism $\varphi:G\to H$ s.t. $\varphi(a)=u$. Moreover, $\varphi$ is injective iff $u$ has infinite order.
	Example: $\aut(\Z_n) \isom U_n$
10/10/2018	Board presentations PS 6
	Thms. 10.2 and 6.3
10/12/2018	Fall break

Week 9	Topics
10/15/2018	Prop. Let N \leq G. TFAE
	(i) $gNg^{-1} \subseteq N$ for all $g\in G$
	(ii) $gNg^{-1} = N$ for all $g\in G$
	(iii) $gN = Ng$ for all $g\in G$
	(iv) the product of any two left cosets is a left coset.
	Moreover, in the last one, we have $(gN)(hN) = ghN$
	Def: normal subgroup
	Examples: 1. $A_n \normaleq S_n$
	$<R_{360/n}> \normaleq D_n$
	Prop: if $H$ is a subgroup of $G$ of index 2, then $H$ is a normal subgroup of $G$
	2. Prop: if $\varphi:G\to \bar{G}$ is a homomorphism, then $ker(\varphi)$ is a normal subgroup of $G$
	3. If $G$ is abelian, then every subgroup of $G$ is normal
	4. $Z(G)$ is a normal subgroup of $G$.
	5. $G$ and $\{1\}$ are normal subgroups of $G$.
	Thm 9.2 proof using (iv) above.
	Example 9.10 Generalize $\Z/n\Z \isom \Z_n$
10/16/2018	Example 9.12
	Thm 10.3 1st Isom Thm
	Example $\varphi:\Z \to \Z_n$
	Thm 9.4
	Thm (N/C theorem) Let $H \leq G$. $N_G(H) / C_G(H)$ is isomorphic to a subgroup of $\aut(H)$.
10/17/2018	proof of N/C theorem
	Example 10.17 $\|G\|=35$
	Thm 9.3
	Corollary: If $\|G\|=pq$ and $Z(G) \neq 1$ then $G$ is abelian.
	Thm 9.5 Cauchy's thm for abelian gps.
10/19/2018	Thm 10.4 $N\normaleq G$, $q:G \to G/N$ is an epimorphism with $ker(q)=N$
	Chapter 8 Direct Product
	Def: $G_1 \oplus G_2$
	Prop: 1) $G_1 \oplus G_2$ is a group.
	2) If $G_1$, $G_2$ are abelian then so is $G_1 \oplus G_2$.

	Examples: (1) $\Z_2 \oplus \Z_3$ is abelian of order 6, so it is isomorphic to $\Z_6$
	(2) $G \oplus \{1\} \isom G \isom \{1\}\oplus G$

	Cor: $G_1 \oplus G_2$ contains subgroups isomorphic to $G_1$ and $G_2$ respectively.
	Def: $G_1 \oplus \cdots \oplus G_n$
	Thm 8.1

Week 10	Topics
10/22/2018	Thm 8.2 $G_1$, $G_2$ finite. $G_1 \oplus G_2$ is cyclic iff $G_1$ and $G_2$ are cyclic or relatively prime orders.
10/23/2018	RSA cryptography. Public vs private keys
	Prop: $m^{ed}\equiv m \pmod n$.
	Internal direct product
	Thm.: Let $H,K\leq G$ be such that $HK=G$ and $H\intersection K=\{1\}$. Then $G\isom H\oplus K$.
	Def: When $H,K\leq G$ are such that $HK=G$ and $H\intersection K=\{1\}$, we say that $G$ is the internal direct product of $H$ and $K$, and write $G=H\times K$.
	Example: Consider $D_n$ with $n=2m$ and $m$ odd.
	Thm. 9.7 and corollary
	Prop: Let $H,N\leq G$.
	(1) If $N\normaleq G$ then $HN\leq G$.
	(2) If $H,N\normaleq G$ then $HN\normaleq G$
10/24/2018	2nd, 3rd, 4th and 5th isomorphism theorems.
	$\sub(D_4)$ and $\sub(V_4)$ as examples.
10/26/2018	Thm If $G$ is a finite abelian group of order $n$, and $m\|n$ then $G$ has a subgroup of order $m$.
	Fund. Thm. of Finite Abelian Groups
	Statement and examples, $n=12$ and $n=600$
	Elementary divisors form, and invariant factors form

Week 11	Topics
10/29/2018	Board presentations. Problems sets 7 and 8
10/30/2018	Ch.24 Def: conjugate, conjugate class $\cl(a)$.
	Prop: (1) “conjugate to” is an equivalence relation. The equivalence classes are the conjugacy classes.
	(2) $\cl(a)=\{a\} \iff a\in Z(G)$
	Thm. 24.1 without finite assumption
	Cor. 1
	Thm. Class equation (2 versions)
	Thm. 24.2 A non-trivial $p$-group has non-trivial center.
	Def: Finite $p$-group. Metabelian group.
	Cor. Let $p$ be a prime. If $\|G\|=p^2$, then $G$ is abelian.
	Cor. Let $p$ be a prime. If $\|G\|=p^3$, then $G$ is metabelian. Moreover, $\|Z(G)\|=p$ or $\|Z(G)\|=p^3$.
	Example: Heisenber group $H$ has order $p^3$, and is not abelian.
10/31/2018	Thm. 24.3 Sylow's 1st Theorem
	Cor. Cauchy's theorem
	Cor. If $\|G\|=pq$ where $p<q$ are primes and $p\not\mid (q-1)$, then $G$ is cyclic.
	Lemma 1. (1) Let $H\leq G$ and $C=\{gHg^{-1}\mid g\in G\}$ the set of all conjugates of $H$. Then $\|C\|=[G:N_G(H)]$.
	Definition of Sylow $p$-subgroup.
	(2) Let $H,K\leq G$. If $HK=KH$ then $HK\leq G$.
	Lemma 2. Let $P$ be a Sylow $p$-subgroup $G$. If $g\in N_G(P)$ and $\|g\|$ is a power of $p$, then $g\in P$.
	Lemma 3. Let $\|G\|=p^km$ and $p\not\mid m$. Let $P$ be a Sylow $p$-subgroup of $G$, i.e. $\|P\|=p^k$, and $H\leq G$ with $\|H\|=p^l$ for some $l\leq k$. Then there is a conjugate of $P$ that contains $H$, i.e. there is $g\in G$ such that $H\leq gPg^{-1}$.
11/02/2018	Board presentations. Problems set 9
	Proof of Lemma3

Week 12	Topics
11/05/2018	Sylow Theorems
	Examples: (1) $\|G\|=35$ (2) $\|G\|=455$ (3) $\|G\|=21$ (4) $\|G\|=256$
11/06/2018	Test 2
11/07/2018	Rings. Definitions: ring, unity, ring with unity (unitary ring), commutative ring, units of a unitary ring
	Examples
	Prop: The units of a ring, $U(R)$ form a multiplicative group.
11/09/2018	No class.

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Daily topics

2018-12-08T18:08:43-04:00

Math 401 - 01 Daily Topics - part 3 (Fall 2018)

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Week 12	Topics
11/05/2018	Sylow Theorems
	Examples: (1) $\|G\|=35$ (2) $\|G\|=455$ (3) $\|G\|=21$ (4) $\|G\|=256$
11/06/2018	Test 2
11/07/2018	Rings. Definitions: ring, unity, ring with unity (unitary ring), commutative ring, units of a unitary ring
	Examples
	Prop: The units of a ring, $U(R)$ form a multiplicative group.
11/09/2018	No class.

Week 13	Topics
11/12/2018	Thm. 12.1
	Thm. 12.2
	Subrings, definition, examples
	Direct Products (Sums), definition, examples
	Ring homomorphisms, definition
	kernel, Ideal
	Homo, mono, epi, iso, endo, auto
11/13/2018	Test 2 returned
	R/I definition
	Thm. 12.3
	Integral Domains, zero-divisors
	Prop. Let $R$ be a commutative ring. TFAE
	(1) $R$ has no zero-divisors
	(2) $R$ satisfies the cancellation law: $ab=ac$ and $a\neq 0 \imp b=c$.
	(3) $R$ satisfies: $ab=0 \imp a=0\ \text{or}\ b=0$
	Definition: integral domain
	Examples: $\Z$, $\Q$, $\R$, $\C$, $\Q(\sqrt{2})$, $\Z_p$.
	Thm. (1) Any field is an integral domain.
	(2) Any finite ID is a field.
	Cor: $\Z_n$ is a field iff it is an ID iff $n$ is a prime.
11/14/2018	Examples: $\Q(\sqrt{2})$ is a field.
	$\Z_3[i]$ is a field.
	$\Z_5[i]$ is not a field.
	Prop: If $R$ is an ID, then $R[x]$ is an ID, and for any $f,g\in R[x]$ we have $\deg(fg)=\deg(f)+\deg(g)$.
	Example: $\Z_6[x]$ is not an ID and the degree formula does not hold.
11/16/2018	Snow day. Class cancelled.

Week 14	Topics
11/19/2018
	(1) $\pbr{a}:=aR=\{ar\|r\in R\}$ is an ideal of $R$.
	(2) $a\in\pbr{a}$.
	(3) If $I\normaleq R$ and $a\in I$ then $\pbr{a} \leq I$.
	Def: $\pbr{a}$ is called the ideal generated by $a$. It is the smallest ideal of $R$ that contains $a$.
	Example: In the ring $\Q[x]/\pbr{x^2-2}$ the element $u=x+I$ where $I=\pbr{x^2-2}$, satisfies $u^2=2$, i.e. it is a root of the polynomial $x^2-2$.
	Characteristic of a ring. Thms. 13.3 and 13.4.
11/20/2018	Comparison of $\Q[\sqrt{2}]$ and $\Q[x]/\pbr{x^2-2}$. Intuitive motivation for the construction $\Q[x]/\pbr{x^2-2}$.
	Given a commutative ring with unity $R$, and an ideal $I\normaleq R$,
	(Q1) when is $R/I$ an I.D.?
	(Q2) when is $R/I$ a field?
	Def: prime ideal
	Thm 14.3. $R/I$ is an ID iff $I$ is a prime ideal.

Week 15	Topics
11/26/2018	Lemma: Let $R$ be a commutative ring with unity, and $I, J\normaleq R$.
	(1) * $I\intersection J \normaleq R$
	* $I\intersection J \leq I, J$
	* if $K\normaleq R$ and $K\leq I, J$ then $K\leq I\intersection J$.
	(2) * $I+J := \{x+y\|x\in I, y\in J\} \normaleq R$
	* $I, J \leq I+J$
	* if $K\normaleq R$ and $I, J\leq K$ then $I+J\leq K$.
	Prop: The set $\idl(R):=\{I\|I\normaleq R\}$ of ideals of $R$ is a lattice, i.e. a partially ordered set, in which any two elements have a $\glb$ and a $\lub$.

	Cor: Let $R$ be a commutative ring with unity and $I\normaleq R$. If $I$ is maximal then it is prime.
11/27/2018	Board presentation, PS 11.
	Example: $R=\Z[x]$, $I=\pbr{x}$ is a prime ideal but it is not maximal.
	Fact: In the ring $\Z$ every ideal is a principal ideal, and every prime ideal is maximal.
11/28/2018	Def: Principal ideal domain (PID).
	Prop: $\Z$ is a principal ideal domain.
	Thm: If $R$ is a PID and $I\normaleq R$ is prime then $I$ is a maximal proper ideal.
	Cor: $\Z[x]$ is not a PID.
	Example: in $\Z[x]$ the ideal $K=\pbr{2}+\pbr{x}$ is not a principal ideal.
	Chapter 15. Divisibility by $9$ criterion.
	Divisibility by $7$ criterion: $n=10m+d_0$ is divisible by $7$ iff $m-2d_0$ is divisible by $7$
11/30/2018	Thm 15.1
	Thm. 15.3
	Lemma: Let $R$ be a ring with unity. For $a,b\in R$, $n,m\in \Z$, $(m\cdot a)(n\cdot b)=(mn)\cdot(ab)$
	Thm. 15.5

Week 16	Topics
12/03/2018	Corollaries 1, 2, and 3.
	Field of fractions(quotients)

12/04/2018	Thm. 15.6 Moreover, $F$ is minimal. If $E$ is a field that contains a copy of $D$, then $E$ contains a copy of $F$.
	Examples. 1) The field of fractions of $\Z$ is $\Q$.
	2) Let $D$ be an integral domain, and $D[x]$ the ring of polynomials over $D$. The field of fractions of $D[x]$ is denoted by $D(x)$, and its elements are called rational functions over $D$. A rational function is a quotient of two polynomials $f(x)/g(x)$, with $g(x)\neq 0$.
	External and internal direct product of groups.
	Def: Internal semi direct product of groups. Given a group $G$, $N\normaleq G$, $H\leq G$ such that $N\intersection H=1$ and $NH=G$, we say that $G$ is the (internal) semi direct product of $N$ and $H$.

12/05/2018	Def: External semi direct product. Given two groups $N$ and $H$ and a homomorphism $\alpha:H\to\aut(N)$, write $\alpha(h)$ as $\alpha_h$. Consider the cartesian product $N\times H$ with the following operation: $$(n_1,h_1)(n_2,h_2)=(n_1\alpha_{h_1}(n_2),h_1h_2)$$
	Thm: 1) The operation just defined makes $N\times H$ into a group. We denote it by $N\rtimes_\alpha H$. We omit the subscript $\alpha$ is it is understood from the context.
	2) $\bar{N}=\{(n,1)\mid n\in N\}$ is a normal subgroup of $N\rtimes_\alpha H$, isomorphic to $N$, via the map $$N\to N\rtimes_\alpha H\\ n\mapsto \bar{n}=(n,1)$$.
	3) $\bar{H}=\{(1,h)\mid h\in H\}$ is a subgroup of $N\rtimes_\alpha H$, isomorphic to $H$, via the map $$H\to N\rtimes_\alpha H\\ h\mapsto \bar{h}=(1,h)$$.
	4) $\bar{N}\intersection \bar{H}=1$ and $\bar{N}\bar{H}=N\rtimes_\alpha H$.
	5) $N\rtimes_alpha H$ is the internal semi direct product of $\bar{N}$ and $\bar{H}$.
	6) Given $h\in H$ and $n\in N$, conjugation of $\bar{n}$ by $\bar{h}$ is given by $$\varphi_{\bar{h}}(\bar{n}) =\overline{\alpha_h(n)}.$$
	Cor: When $\alpha$ is the trivial homomorphism, i.e. $\alpha_h=1$ for all $h\in H$, then the semi direct product is equal to the direct product, $N\rtimes_\alpha H=N\oplus H$.
	Cor: The operation in $N\rtimes_\alpha H$ is completely determined by the operations in $N$ and $H$, and the relation $$\bar{h}\ \bar{n}=\overline{\alpha_h(n)}\ \bar{h}.$$
	Example: Let $N=\pbr{a}$ be cyclic of order $7$, and $H=\pbr{b}$ cyclic of order $3$. $\aut(N)\isom U_7$ is abelian of order $6$, hence cyclic.
	$s:N\to N, a\mapsto a^2$ is an automorphism of $N$ of order $3$ since $a^{2^3}=a^8=a$.
	$\aut(N)$ is generated by $c:N\to N, a\mapsto a^3$, and $s=c^2$, since $a^{3^2}=a^9=a^2$.
	Any homomorphism $\alpha:H\to\aut(N)$ has to map $b$, which has order $3$, to an element of $\aut(N)$ of order a divisor of $3$. The only such elements are $1$, $s$ and $s^{-1}=c^4=s^2$.
12/07/2018	Board presentation PS 12
	Continuation of example. There are three different semi direct product of $N$ and $H$, given by the three automorphisms $\alpha(b)=1$, $\beta(b)=s$, and $\gamma(b)=s^2$. Let's write down the three.
	Case 1: $\alpha(b)=1$ is trivial. In this case $N\rtimes_\alpha H = N\oplus H\isom C_{21}$.
	Case 2: $\beta(b)=s$. $$ba = \beta_b(a)b = s(a)b = a^2b$$
	so $N\rtimes_\beta H$ is not abelian, and not isomorphic to case 1.
	Case 3: To distinguish from case 2, let's write $N=\pbr{u}$ cyclic of order $7$, and $H=\pbr{v}$ cyclic of order $3$, $\gamma(v)=s^2$. $$vu = \gamma_v(u)v = s^2(u)v = u^4v$$
	Again $N\rtimes_\gamma H$ is not abelian, not isomorphic to case 1.
	Claim: $N\rtimes_\beta H$ is isomorphic to $N\rtimes_\gamma H$ via the map $a\mapsto u, b\mapsto v^{-1}$.
	Cor: there are only two non-isomorphic semi direct products of a cyclic group of order 7 and a cyclic group of order 3, namely, the direct product, and the non-abelian semi direct product of case 2.
	Example: Let $G$ be a group of order $21$. By Sylow's theorem we have $n_7=1$. Let $N$ be the Sylow 7-subgroup of $G$. We also know that $n_3$ is either 1 or 7. Let $H$ be a Sylow 3-subgroup of $G$. When $n_3=1$ $H$ is a normal subgroup of $G$ and $G$ is the direct product of $N$ and $H$. When $n_3=7$, then $H$ is not normal, and $G$ is the non-abelian semi direct product of $N$ and $H$.
	Therefore, there are exactly two non-isomorphic groups of order 21.

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Modern Algebra I - Math 401 - 01 (Fall 2018)

2018-12-07T15:58:34-04:00

Modern Algebra I - Math 401 - 01 (Fall 2018)

Fall 2018

Syllabus

Instructor:	Fernando Guzmán	WH-116	x-72876	fer@math.binghamton.edu

Classroom:	AP-G15	MWF	9:40 - 10:40
Classroom:	OR-100D	T	10:05 - 11:30

Office Hours:	Monday	1:15 - 2:15	p.m.
(subject to change)	Tuesday	4:00 - 5:00	p.m.
(subject to change)	Friday	8:30 - 9:30	a.m.

Announcements

Office hours during finals week:
Mon 2:30-3:30
Wed 9:30-10:30
Fri 10:00-11:00

Review session: Monday 12/10/2018, 3:30 - 4:30 pm in WH 309

Final Exam in Tuesday 12/11/2018, 12:50 - 2:50 pm in AAG 007

Homework

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Homework

2018-12-07T14:11:31-04:00

Math 401 - 01 Homework (Fall 2018)

Problem Set 13 (complete) Due: 12/10/2018, optional (bring to review session)

Let $D$ be an I.D., $D[x]$ the ring of polynomials over $D$, and $D(x)$ the field of rational functions over $D$. Let $F$ be the field of fractions of $D$, $F[x]$ the ring of polynomials over $F$, and $F(x)$ the field of rational functions over $F$. Show that $D(x)=F(x)$.
Prove that the operation that defines the external semi-direct product is in fact associative.
Prove that the two non-abelian semi-direct products of $C_7$ with $C_3$ are isomorphic. (Hint: use the homomorphism discussed in class, given by: $a\mapsto u, b\mapsto v^{-1}$)

Problem Set 12 (complete) Due: 12/03/2018. Board presentation: 12/07/2018

Chapter 14, problems 12, 14. Warning: pay attention to the definition of $AB$.
Chapter 14, problem 28. What about the converse?
Write $n\in\Z$ as $md_0$, where $d_0$ is the last digit (base 10) and $m$ consists of all other digits. In other words, $n=10 m+d_0$. Prove that $n$ is divisible by $7$ iff $m-2d_0$ is divisible by $7$.

Problem Set 11 (complete) Due: 11/20/2018. Board presentation: 11/27/2018

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

Previous Homework

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Previous Homework

2018-11-27T16:41:20-04:00

Math 401 - 01 Previous Homework (Fall 2018)

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

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 $HK\leq G$.
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:
1. If $N\normaleq G$, then $HN\leq G$.
2. 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 $60$.
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,

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