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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}}
+[[people:fer:401ws:fall2018:home| Home]]
+^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  $G_1 \oplus G_2$ .|
+| |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>&nbsp;&nbsp;</html> (2)  $|G|=455$  <html>&nbsp;&nbsp;</html> (3)  $|G|=21$  <html>&nbsp;&nbsp;</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.|
+[[people:fer:401ws:fall2018:daily_topics_3|Daily topics (3)]]
+[[people:fer:401ws:fall2018:home| Home]]

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