{"id":6748,"date":"2016-02-01T15:24:58","date_gmt":"2016-02-01T19:24:58","guid":{"rendered":"http:\/\/nycphantom.com\/journal\/?p=6748"},"modified":"2016-05-25T09:37:08","modified_gmt":"2016-05-25T13:37:08","slug":"the-city-college-cuny-class-math-34700-modern-algebra","status":"publish","type":"post","link":"http:\/\/nycphantom.com\/journal\/?p=6748","title":{"rendered":"The City College CUNY Class: MATH 34700 Modern Algebra"},"content":{"rendered":"<p>Professor: Benjamin Steinberg<\/p>\n<p><a href=\"http:\/\/math.sci.ccny.cuny.edu\/pages?name=Math+34700\" target=\"_blank\">Syllabus<\/a><\/p>\n<p>HW must be from 8th edition of textbook.<\/p>\n<p>Set-builder notation.<\/p>\n<p>Empty set is contained in all sets.<\/p>\n<p>A = B, if <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_14e818fb2e89b04d5fe875ac9ae0199b.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p>{a,b,c} = {a,a,b,b,c,c,c}<\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Theorems and Definitions of the text: (variables are generally assumed to be integers unless stated otherwise)<\/strong><\/span><\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Chapter 1 Fundamentals<\/strong><\/span><\/p>\n<p><strong>Section 1.1 Sets<\/strong><\/p>\n<p><strong>Section 1.2 Mappings<\/strong><\/p>\n<p><strong>Section 1.3 Properties of Composite Mappings (Optional)<\/strong><\/p>\n<p><strong>Section 1.4 Binary Operations<\/strong><\/p>\n<p><strong>Section 1.5 Permutations and Inverses<\/strong><\/p>\n<p><strong>Section 1.6 Matrices<\/strong><\/p>\n<p><strong>Section 1.7 Relations<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Chapter 2 The Integers<\/strong><\/span><\/p>\n<p><strong>Section 2.1 Postulates for the Integers (Optional)<\/strong><\/p>\n<p><strong>Section 2.2 Mathematical Induction<\/strong><\/p>\n<p><strong>Section 2.3 Divisibility<\/strong><\/p>\n<p><strong>Section 2.4 Prime Factors and GCD<br \/>\n<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 2.11<\/span> Greatest Common Divisor: gcd(a,b) = d<br \/>\n1. d &gt; 0<br \/>\n2. d|a &amp; d|b<br \/>\n3. c|a &amp; c|b =&gt; c|d<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 2.12<\/span> Greatest Common Divisor<br \/>\nA method to find gcd: Euclidean Algorithm<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 2.13<\/span> Relatively Prime Integers: gcd(a,b) = 1<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 2.14<\/span> If gcd(a,b) = 1 &amp; a|bc =&gt; a|c<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 2.15<\/span> Prime Integer (p)<br \/>\n1. p &gt; 1<br \/>\n2. the only divisors of p: \u00b1 1 and \u00b1p<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 2.16<\/span> Euclid's Lemma<br \/>\nprime p |ab,<br \/>\n=&gt; either p|a or p|b<\/p>\n<p><span style=\"text-decoration: underline;\">Corollary 2.17<\/span> prime p|<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_8192f5fb63ad50618efa7466886515ab.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 2.18<\/span> Unique Factorization Theorem = Fundamental Theorem of Arithmetic<br \/>\nn &gt; 0,<br \/>\n=&gt; n=1 or <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_3c4b44fdc86ba2cabf0276e2f76b4ef9.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>..., where <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_37776ef04e8a6a01d5ffe1dcf1322214.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> are unique primes<br \/>\nTerms: <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_342e772474b691ac87dac30aeef596c0.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> = <strong>multiplicity<\/strong> of <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_eca91c83a74a2373ca5f796700e99fd3.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> =&gt; <strong>standard form<\/strong> for <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_f18cd49b55e427bddf0b86ccd9468877.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>...<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 2.19<\/span> Least Common Multiple: lcm(a,b) = m where a, b != 0<br \/>\n1. m &gt; 0<br \/>\n2. a|m &amp; b|m<br \/>\n3. a|c &amp; b|c =&gt; m|c<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 2.20<\/span> Euclid's Theorem: The number of primes is infinite<\/p>\n<p><strong>Section 2.5 Congruence of Integers<br \/>\n<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 2.21<\/span> Congruence modulo n<br \/>\nn &gt; 1, x \u2261 y (mod n)<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 2.22<\/span> Congruence modulo n is an equivalence relation on <strong>Z<\/strong>.<br \/>\nCongruence classes = residue classes<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 2.23<\/span> Addition and Multiplication Properties<br \/>\na \u2261 b (mod n),<br \/>\n=&gt; a+x \u2261 b+x (mod n) &amp; ax \u2261 bx (mod n)<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 2.24<\/span> Substitution Properties<br \/>\na \u2261 b (mod n) &amp; c \u2261 d (mod n),<br \/>\n=&gt; a+c \u2261 b+d (mod n) &amp; ac \u2261 bd (mod n)<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 2.25<\/span> Cancellation Law<br \/>\ngcd(a,n) = 1 &amp; ax \u2261 ay (mod n),<br \/>\n=&gt; x \u2261 y (mod n)<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 2.26<\/span> Linear Congruences<br \/>\ngcd(a,n) = 1,<br \/>\n\u2203x =&gt;ax = b (mod n)<br \/>\n=&gt; <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_6c5ad148268c5754bfb6c356fa04caa3.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> (mod n)<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 2.27<\/span> System of Congruences<br \/>\ngcd(m,n) = 1,<br \/>\n\u2203x =&gt; x \u2261 a (mod m), x \u2261 b (mod n),<br \/>\n=&gt; <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_6c5ad148268c5754bfb6c356fa04caa3.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> (mod mn)<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 2.28<\/span> Chinese Remainder Theorem<br \/>\nexpands theorem 2.27<\/p>\n<p><strong>Section 2.6 Congruence Classes: <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_c4181b39fa51ad9a30657febc31540b2.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> = {[0],[1],...,[n-1]}<br \/>\n<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 2.29<\/span> Addition in <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_c4181b39fa51ad9a30657febc31540b2.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>: [a] + [b] = [a+b]<br \/>\n1. Associative &amp; commutative in <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_c4181b39fa51ad9a30657febc31540b2.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\n2. [0] = additive identity<br \/>\n3. [-a] is additive inverse of [a], where [-a],[a] <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_fce07b650ef0f81b404f9a179ea9ca28.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 2.30<\/span> Multiplication in <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_c4181b39fa51ad9a30657febc31540b2.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>: [a][b] = [ab]<br \/>\n1. Associative &amp; Commutative in <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_c4181b39fa51ad9a30657febc31540b2.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\n2. [1] = multiplicative identity<br \/>\n3. <span style=\"text-decoration: underline;\">Theorem 2.31<\/span> [a] <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_fce07b650ef0f81b404f9a179ea9ca28.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> has multiplicative inverse in <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_c4181b39fa51ad9a30657febc31540b2.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> IFF gcd(a,n) = 1<\/p>\n<p><span style=\"text-decoration: underline;\">Corollary 2.32<\/span> [a] in <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_c4181b39fa51ad9a30657febc31540b2.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> has multiplicative inverse IFF n is a prime<\/p>\n<p><strong>Section 2.7 Introduction to Coding Theory (Optional) - skipped<\/strong><\/p>\n<p><strong>Section 2.8 Introduction to Cryptography (Optional)<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 2.33<\/span> RSA Public Key Cryptosystem<\/p>\n<p><strong>Chapter 3 Groups<\/strong><\/p>\n<p><strong>Section 3.1 Definition of Group<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 3.1<\/span> Group, G<br \/>\n1. G is Closed under *<br \/>\n2. * is associative<br \/>\n3. G has identity element e<br \/>\n4. all elements of G has inverses in G<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 3.2<\/span> Abelian Group (=Commutative group)<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 3.3<\/span> Finite, Infinite groups, Order of Group (|G|)<\/p>\n<p><strong>Section 3.2 Properties of Group Elements<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 3.4<\/span> Properties of Group Elements<br \/>\n1. e in G is unique<br \/>\n2. All inverses of elements in G are in G and unique<br \/>\n3. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_c0c5e33e3cfc20c7284408a4a6859002.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\n4. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_248805f2e74a487e175d7ba128f84d84.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> Reverse order law<br \/>\n5. ax = ay or xa = ya imples x = y: Cancellation law<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 3.5<\/span> Equivalent Conditions for a Group<br \/>\nG is nonempty set, closed under associative multiplication.<br \/>\nG is a group IFF ax = b and ya = b s.t. x and y in G, for all a and b in G.<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 3.6<\/span> Product Notation<br \/>\n<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_81e22b84f5c5b72db7b05288c582eadd.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> for <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_55d8f2783bde156673a0eed19e339a37.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 3.7<\/span> Generalized Associative Law<br \/>\n<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_b7c87a9fe844246fcf52b08505e5c6e4.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p>Definition 3.8 Integral Exponents<br \/>\nCompare Multiplicative and Additive Notations<\/p>\n<p>Theorem 3.9 Laws of Exponents<br \/>\n1. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_be4752b7ef689fbb02f00a959f510659.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\n2. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_73c179b4ab706a0524982210be6825b4.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\n3. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_9920b25dadc6795f7ab6270b19bd76d5.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\n4. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_0212bdd042b14974bd59b51830864994.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\nCan be translated to Laws of Multiples for additive groups.<\/p>\n<p><strong>Section 3.3 Subgroups<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 3.10<br \/>\nTheorem 3.11<\/span> Subgroup,\u00a0H <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_b92481eddde8c0a762bc2eab35a80a37.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> G is a subgroup IFF:<br \/>\n1.) identity e is in H (H is nonempty)<br \/>\n2.) H is closed under *<br \/>\n3.) <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_aec0642faa0a8f2ab1771ca10568e4c0.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>, closed under inverse<br \/>\nNotation: <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_651c6b8e2e1dfea84d2b84e33f91ee84.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> is a subgroup of G<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 3.12<\/span> Equivalent Set of Conditions for a Subgroup (Summarized Theorem 3.11)<br \/>\nH is subgroup of IFF:<br \/>\n1. H is nonempty (I think this can be ignored)<br \/>\n2. a, b in H implies <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_4e3e93201550469ed49c21742c1a23ac.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 3.13<\/span> The center of a Group<br \/>\nZ(G) = {<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_48abf8e0ca92405b5c2499e862f07c7b.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>}<br \/>\nMeaning all g which commute with every element of G<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 3.14<\/span> The center of a group G is an abelian subgroup of G<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 3.15<\/span> The Centralizer of a Group Element<br \/>\n<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_98ec9983a7068d30e34946b31a46e303.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 3.16<\/span> The centralizer of a in G is a subgroup of G<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 3.17<\/span> Cyclic Subgroup<br \/>\nFor any a in group G, &lt;a&gt; = subgroup generated by a = <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_9c5a5354c51b421ee970db75e447ce6c.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> = H = cyclic subgroup of G. The element a is called a <strong>generator<\/strong> of H.<\/p>\n<p><strong>Section 3.4 Cyclic Groups<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 3.18<\/span> Generator (repeated from last section)<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 3.19<\/span> Infinite Cyclic Group<br \/>\nIf <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_f04420b1ba8643d36546f4e1109d9d0f.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> whenever <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_9614351a06bf2e321b2aae2a1266810e.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> and &lt;a&gt; is an infinite cyclic group<\/p>\n<p><span style=\"text-decoration: underline;\">Corollary 3.20<\/span> If G is a finite group, then <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_40afc77505ecf9effd1eb393217ddd44.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> for some positive integer n.<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 3.21<\/span> Finite Cyclic Group<br \/>\nIf m is the least positive integer s.t. a^m = e, then<br \/>\n1. &lt;a&gt; has order m<br \/>\n2. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_4a32e9f95d8841dfd1a9240090ede97b.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> IFF <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_0a790ffceae4f86d1baf96640878df13.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> (mod m)<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 3.22<\/span> Order of an Element: o(a) = |a| = |&lt;a&gt;|<br \/>\nOrder of Subgroup of G generated by a in G<\/p>\n<p><span style=\"text-decoration: underline;\">Corollary 3.23<\/span> Finite Order of an Element<br \/>\nIf |a| is finite, then m = |a| is the least positive integer s.t. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_6715451be05aca953947dbda1f283c19.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 3.24<\/span> Subgroup of a Cyclic Group<br \/>\nIf H is a subgroup of cyclic group G with generator a, then<br \/>\n<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_3cca27d1ebfcf11ea8b5addf1279b5b6.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>, where k is the least positive integer s.t. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_90f64a4ac1ac277f8b03f161cd198d05.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Corollary 3.25<\/span> Any subgroup of a cyclic group is cyclic<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 3.26<\/span> Generators of Subgroups<br \/>\nSubgroup (generated by\u00a0<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_da793f82227a193582db78488bd082fa.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>) of cyclic group of order n generated by a = subgroup generated by <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_1f76ba398de2d84074e2c2c040e922c2.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>, where d = gcd(m,n)<\/p>\n<p><span style=\"text-decoration: underline;\">Corollary 3.27<\/span> Distinct Subgroups <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_6191db5db8655090ad54745f2e77daab.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> of a Finite Cyclic Group, G<br \/>\nwhere d is a positive divisor of n.<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 3.28<\/span> Generators of a Finite Cyclic Group &lt;a&gt;:<br \/>\n<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_da793f82227a193582db78488bd082fa.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> is a generator of &lt;a&gt; of order n IFF gcd(m,n) = 1<\/p>\n<p><strong>Section 3.5 Isomorphisms<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 3.29<\/span> Isomorphism, Automorphism<br \/>\n<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_ea4cc14e93ee10eb4ce01de01290337d.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> is an isomorphism from G to G' if:<br \/>\n1. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_87567e37a1fe699fe1c5d3a79325da6f.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> is a bijection from G to G'<br \/>\n2. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_7e9d13097d83781c575627929265210b.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\nNotation <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_dce8f9044c69088eaba61921e4e55ae3.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\nAutomorphism: An isomorphism from G to G<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 3.30<\/span> Images of Identities and Inverses<br \/>\n1. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_65a3ef25769a173fcc35eb014e0f0352.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> and<br \/>\n2. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_6d306ced4c3b5e56b8366069fa9db186.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><strong>Section 3.6 Homomorphisms<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 3.31<\/span> Groups G,<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_497a075274720085438084843329c6d0.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> &amp; G'',<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_6718f3851aba6f5727da3da31bc74ee2.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>, <strong>Homomorphism<\/strong>: <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_ead26ea408ed98341c8da779523673ff.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> s.t. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_51b62f5d81d7ccd633b689743b2f888f.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> or<br \/>\n<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_b429f27333410f7785de00e2f46fc97c.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\n1. <strong>Endomorphism<\/strong>: G = G'<br \/>\n2. <strong>Epimorphism<\/strong>: <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_87567e37a1fe699fe1c5d3a79325da6f.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> is onto, G' is <em>homomorphic image<\/em> of G.<br \/>\n3. <strong>Monomorphism<\/strong>: <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_87567e37a1fe699fe1c5d3a79325da6f.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> is one-to-one<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 3.32<\/span> Images of Identities and Inverses (<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_0c721d7b0746a8c2045a532f05ce2795.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>)<br \/>\n1. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_65a3ef25769a173fcc35eb014e0f0352.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\n2. <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_6d306ced4c3b5e56b8366069fa9db186.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 3.33<\/span> Kernel: ker <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_6e6b921c40bb48a59d3d0183adf3fa2d.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\n<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_87567e37a1fe699fe1c5d3a79325da6f.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> is a homomorphism from group G to G'<\/p>\n<p><strong>Chapter 4 More on Groups<\/strong><\/p>\n<p><strong>Section 4.1 Finite Permutation Groups<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 4.1<\/span> Cycle<br \/>\n<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_53affb4eb822549961ed7e34e6db8969.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\nThis statement is not unique.<br \/>\nDisjoint subsets which define cycles, are called orbits<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 4.2<\/span> Partition of A<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 4.3<\/span> Orbit<br \/>\nLet <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_b9263a71e66eb792fbaf5b3912a321fb.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>, the orbit of <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_221c794bcd277065db1af5dbbc13e502.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\nThe orbits form a partition of {1,...,n}<br \/>\n<strong>Note<\/strong>: All permutations can be written as a product of transpositions:<br \/>\n<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_0e181917345daf47d3423537cfce286e.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Lemma 4.4<\/span> t(P) = (-1)P<br \/>\nt is any transposition (r,s) on {1,2,...,n} and <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_583f2ac8dd240286815657f746ca2321.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 4.5<\/span> Products of Transpositions (Either all even or all odd transpositions)<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 4.6<\/span> Even Permutation vs. Odd Permutation (based on number of transpositions)<br \/>\nEven Permutation if r in r-cycle is odd<br \/>\nOdd Permutation if r in r-cycle is even<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 4.7<\/span> Alternating Group <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_f93809ae14fb28ef6dbe11c99529c51b.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\nis the <strong>subgroup<\/strong> of <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_88e99f0b764d313c50a5f4fdd8a7947e.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> <strong>that consists<\/strong> all even permutations in <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_88e99f0b764d313c50a5f4fdd8a7947e.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 4.8<\/span> Conjugate Elements<br \/>\n1. The <strong>Conjugate<\/strong> of a by b is the element <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_78a4fccac6a99b4b81a6930a294810c1.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\n2. c is a <strong>conjugate<\/strong> of a IFF <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_79d802757a24035178d5aea9ae184862.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> for some b in G<\/p>\n<p><strong>Section 4.2 Cayley's Theorem<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 4.9<\/span> Cayley's Theorem<br \/>\nEvery group is isomorphic to a group of permutations.<\/p>\n<p><strong>Section 4.4 Cosets of a Subgroup<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 4.10<\/span> Product of Subsets<br \/>\nLet <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_9c2bd4bf8f58d987faf0a961abca317e.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 4.11<\/span> Properties of the Product of Subsets<br \/>\n1. Associative<br \/>\n2. not commutative<br \/>\n3. B = C <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_055889aaee38b7c53f994c5e42a40994.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> AB = AC &amp; BA = CA<br \/>\n4. AB = AC\u00a0<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_947bc4afe4d6144fcee59fe7b91aa93c.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> B = C<br \/>\n5. gA = gB <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_055889aaee38b7c53f994c5e42a40994.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> A = B<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 4.12<\/span> Cosets,\u00a0H &lt; G, <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_4ccc9b4493c9cba0d7dad1d836e0c182.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><br \/>\nLeft coset of H in G: aH = {<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_ebf2d86bf8116340fad9bbe19110c5be.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> | x = ah for some <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_1b499f2a7a708ad6a2c28505c128c5af.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>}<br \/>\nRight coset of H in G: Ha<br \/>\nLeft &amp; right cosets are never disjoint, but they may be different sets.<\/p>\n<p><span style=\"text-decoration: underline;\">Lemma 4.13<\/span> Left Coset Partition<br \/>\nThe distinct left cosets of H form a partition of G.<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 4.14<\/span> Index: [G:H]<br \/>\nThe total number of distinct left cosets of H in G.<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 4.15<\/span> Lagrange's Theorem<br \/>\nIf G is a finite group, <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_7052806b2d025ffdf8e8ef749d6354cb.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Corollary 4.16<\/span> |a| divides |G|<\/p>\n<p><strong>Section 4.5 Normal Subgroups<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 4.17<\/span> Normal (Invariant) Subgroup<br \/>\nH is a normal subgroup of G if xH = Hx for all x in G (left coset = right coset)<br \/>\nNote that it does not mean xh = hx<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 4.18<\/span> A special coset H: hH = Hh = H<\/p>\n<p><span style=\"text-decoration: underline;\">Corollary 4.19<\/span> The square of a subgroup <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_745a2d7154cc1621a45e87c3caeddbed.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 4.20<\/span> Normal Subgroups and Conjugates<br \/>\nH is a normal subgroup of G IFF <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_a6e0588e7bcb46ba7a43f8d16ee2449c.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 4.21<\/span> Set Generated by A<br \/>\n<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_f412279fecb91d8a89ac13f70a408acb.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 4.22<\/span> Subgroup Generated by A: &lt;A&gt; is a subgroup of G generated by A<\/p>\n<p><strong>Section 4.6 Quotient Groups<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 4.23<\/span> Group of Cosets<br \/>\nIf H is normal subgroup of G, then cosets of H form a group with respect to the product of subsets<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 4.24<\/span> Quotient (Factor) Group of G by H<br \/>\nIf H is normal subgroup of G, the group G\/H consists of the cosets of H in G is the quotient group of G by H<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 4.25<\/span> Quotient Group =&gt; Homomorphic Image<br \/>\n<span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_87567e37a1fe699fe1c5d3a79325da6f.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>(a) = aH is an epimorphism from G to G\/H<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 4.26<\/span> Kernel of a Homomorphism<br \/>\nIf f is a homomorphism from G to G', ker f is a normal subgroup of G<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 4.27<\/span> Homomorphic Image =&gt; Quotient Group<br \/>\nLet G and G' be groups with G' a homomorphic image of G. Then G' is isomorphic to aquotient group of G<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 4.28<\/span> Fundamental Theorem of Homomorphisms<br \/>\nIf <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_87567e37a1fe699fe1c5d3a79325da6f.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> is an epimorphism from the group G to the group G', then G' is isomorphic to G\/ker <span class='MathJax_Preview'><img src='http:\/\/nycphantom.com\/journal\/wp-content\/plugins\/latex\/cache\/tex_87567e37a1fe699fe1c5d3a79325da6f.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script><\/p>\n<p><strong>Skipped<\/strong><\/p>\n<p><strong>Chapter 5 Rings, Integral Domains and Fields<\/strong><\/p>\n<p><strong>Section 5.1 Definition of a Ring<\/strong><\/p>\n<p><span style=\"text-decoration: underline;\">Definition 5.1a<\/span> A set R is a ring it has all 6 properties:<br \/>\na) closed under + and .<br \/>\nb) associative under + and .<br \/>\nc) commutative under +<br \/>\nd) contains additive identity = zero = 0<br \/>\ne) contains additive inverses = -a = negative (opposite) of a, applies in subtraction<br \/>\nf) two distributive laws hold<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 5.1b<\/span> Alternative Definition of a Ring:<br \/>\na) R forms an abelian group under +<br \/>\nb) R is closed under associative multiplication<br \/>\nc) two distributive laws hold<\/p>\n<p><span style=\"text-decoration: underline;\">Definition 5.2<\/span> Subring, analogous to subgroup<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 5.3<\/span> Equivalent set of conditions for a subring<br \/>\na) S is nonempty<br \/>\nb) x, y in S implies x+y and xy are in S<br \/>\nc) x in S implies -x in S<\/p>\n<p><span style=\"text-decoration: underline;\">Theorem 5.4<\/span> Characterization of a subring<br \/>\na) S is nonempty<br \/>\nb) x, y in S implies x-y and xy are in S<br \/>\nS = {0} and S = R are trivial subrings<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Professor: Benjamin Steinberg Syllabus HW must be from 8th edition of textbook. Set-builder notation. Empty set is contained in all sets. A = B, if {a,b,c} = {a,a,b,b,c,c,c} Theorems and Definitions of the text: (variables are generally assumed to be &hellip; <a href=\"http:\/\/nycphantom.com\/journal\/?p=6748\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[27,10],"tags":[],"class_list":["post-6748","post","type-post","status-publish","format-standard","hentry","category-mathematics","category-projects"],"_links":{"self":[{"href":"http:\/\/nycphantom.com\/journal\/index.php?rest_route=\/wp\/v2\/posts\/6748","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/nycphantom.com\/journal\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/nycphantom.com\/journal\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/nycphantom.com\/journal\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/nycphantom.com\/journal\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=6748"}],"version-history":[{"count":52,"href":"http:\/\/nycphantom.com\/journal\/index.php?rest_route=\/wp\/v2\/posts\/6748\/revisions"}],"predecessor-version":[{"id":6990,"href":"http:\/\/nycphantom.com\/journal\/index.php?rest_route=\/wp\/v2\/posts\/6748\/revisions\/6990"}],"wp:attachment":[{"href":"http:\/\/nycphantom.com\/journal\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6748"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/nycphantom.com\/journal\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6748"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/nycphantom.com\/journal\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6748"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}