Problem of the Week
Math Club
BUGCAT 2020
Zassenhaus Conference
Hilton Memorial Lecture
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Week 12 | Topics |
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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 |
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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 |
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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 |
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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 |
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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. |