~~META:title=Daily topics~~
===== 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 = $ 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. $A_n \normaleq S_n$|
| | $ \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