people:fer:401ws:fall2018:daily_topics
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| + | ~~META:title=Daily topics~~ | ||
| + | <WRAP centeralign> | ||
| + | ===== Math 401 - 01 Daily Topics - part 2 (Fall 2018)===== | ||
| + | </WRAP> | ||
| + | ---- | ||
| + | {{page>people:fer:401ws:defs&nofooter&noeditbtn}} | ||
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| + | [[people:fer:401ws:fall2018:home| Home]] | ||
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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| | ||
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| + | ^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 $G_1 \oplus G_2$.| | ||
| + | | |3) If $G_1$, $G_2$ are finite then so is $G_1 \oplus G_2$ and $$| G_1 \oplus G_2| = |G_1|\cdot |G_2|$$ | ||
| + | | |Examples: (1) $\Z_2 \oplus \Z_3$ is abelian of order 6, so it is isomorphic to $\Z_6$| | ||
| + | | |(2) $G \oplus \{1\} \isom G \isom \{1\}\oplus G$| | ||
| + | | |Prop: Let $H_1 \leq G_1$ and $H_2 \leq G_2$. Then $$H_1 \oplus H_2 \leq G_1 \oplus G_2$$ | ||
| + | | |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) $|G|=35$ <html>  </html> (2) $|G|=455$ <html>  </html> (3) $|G|=21$ <html>  </html> (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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| + | [[people:fer:401ws:fall2018:daily_topics_3|Daily topics (3)]] | ||
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| + | [[people:fer:401ws:fall2018:home| Home]] | ||