Daily topics [Department of Mathematics and Statistics, Binghamton University]

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 $G = <a>$ 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 $G_1 \oplus G_2$ .

	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 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) $\|G\|=35$ (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.

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

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