# ### Site Tools

people:fer:401ws:fall2018:daily_topics

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

$\newcommand{\aut}{\textrm{Aut}} \newcommand{\inn}{\textrm{Inn}} \newcommand{\sub}{\textrm{Sub}} \newcommand{\cl}{\textrm{cl}} \newcommand{\join}{\vee} \newcommand{\bigjoin}{\bigvee} \newcommand{\meet}{\wedge} \newcommand{\bigmeet}{\bigwedge} \newcommand{\normaleq}{\unlhd} \newcommand{\normal}{\lhd} \newcommand{\union}{\cup} \newcommand{\intersection}{\cap} \newcommand{\bigunion}{\bigcup} \newcommand{\bigintersection}{\bigcap} \newcommand{\sq}[\ ]{\sqrt[#1]{#2\,}} \newcommand{\pbr}{\langle #1\rangle} \newcommand{\ds}{\displaystyle} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\A}{\mathbb{A}} \newcommand{\F}{\mathbb{F}} \newcommand{\T}{\mathbb{T}} \newcommand{\ol}{\overline{#1}} \newcommand{\imp}{\Rightarrow} \newcommand{\rimp}{\Leftarrow} \newcommand{\pinfty}{1/p^\infty} \newcommand{\power}{\mathcal{P}} \newcommand{\calL}{\mathcal{L}} \newcommand{\calC}{\mathcal{C}} \newcommand{\calN}{\mathcal{N}} \newcommand{\calB}{\mathcal{B}} \newcommand{\calF}{\mathcal{F}} \newcommand{\calR}{\mathcal{R}} \newcommand{\calS}{\mathcal{S}} \newcommand{\calU}{\mathcal{U}} \newcommand{\calT}{\mathcal{T}} \newcommand{\gal}{\textrm{Gal}} \newcommand{\isom}{\approx} \newcommand{\idl}{\textrm{Idl}} \newcommand{\lub}{\textrm{lub}} \newcommand{\glb}{\textrm{glb}}$

Week 7Topics
10/01/2018Test 1
10/02/2018Lagrange's corollary 1
Orbit-Stabilizer theorem
Examples: cube, truncated icosahedron (soccer ball)
10/03/2018Corollaries 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)
10/05/2018Thm. 7.3
Week 8Topics
10/08/2018Test 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
$\aut(G)$, $\inn(G)$
10/09/2018Thm 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: $\aut(\Z_n) \isom U_n$
10/10/2018Board presentations PS 6
Thms. 10.2 and 6.3
10/12/2018Fall break
Week 9Topics
10/15/2018Prop. 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.
10/19/2018Thm 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 10Topics
10/22/2018Thm 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.
10/23/2018RSA 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 $HN\normaleq G$
10/24/20182nd, 3rd, 4th and 5th isomorphism theorems.
$\sub(D_4)$ and $\sub(V_4)$ as examples.
10/26/2018Thm 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 11Topics
10/29/2018Board presentations. Problems sets 7 and 8
10/30/2018Ch.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.
10/31/2018Thm. 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 $H\leq gPg^{-1}$.
11/02/2018Board presentations. Problems set 9
Proof of Lemma3
Week 12Topics
11/05/2018Sylow Theorems
Examples: (1) $|G|=35$    (2) $|G|=455$    (3) $|G|=21$    (4) $|G|=256$
11/06/2018Test 2
11/07/2018Rings. 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.
11/09/2018No class. 