[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571859-105":53,"doc-detail-571859-en":127},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","harolds-abstract-algebra-cheat-sheet-draft","Harold’s Abstract Algebra Cheat Sheet - Draft","","Harold’s Abstract Algebra Cheat Sheet (DRAFT, 5 May 2026) compiles core symbols and foundational results used throughout introductory abstract algebra. It defines common number systems and operations (sets, inclusions, inequalities, inverses, mod arithmetic), then summarizes key theorems and corollaries including division algorithm, gcd and lcm properties, Euclid’s lemma, and the fundamental theorem of arithmetic. It also includes principles of mathematical induction, DeMoivre’s theorem, complex-number properties, and the structure of equivalence relations and equivalence classes.",{"@graph":63,"@context":119},[64,80,102],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/harolds-abstract-algebra-cheat-sheet-draft/571859/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/harolds-abstract-algebra-cheat-sheet-draft/571859.png","ImageObject",442,249,{"name":88,"@type":89},"Miles","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-25","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",7,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What does the cheat sheet cover under “Symbols”?,","Question",{"text":109,"@type":110},"It lists standard mathematical symbols and short definitions covering empty sets and number systems (ℕ, ℤ, ℚ, ℝ, ℂ), subset and element relations, modular arithmetic, inverses, inequalities, and group-related notation such as order.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"Which core number-theory results are included in “Ch. 0: Preliminaries”?,",{"text":114,"@type":110},"It includes the Division Algorithm, statements and consequences around gcd (including a linear combination form), Euclid’s Lemma, and the Fundamental Theorem of Arithmetic, plus LCM-related definitions and examples.",{"name":116,"@type":107,"acceptedAnswer":117},"What algebra topics beyond number theory are included?,",{"text":118,"@type":110},"It summarizes properties of complex numbers, DeMoivre’s theorem, the first and second principles of mathematical induction, and the definition of equivalence relations and their partitioning behavior.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},571859,1792005605,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":136,"language":137,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":138,"faqs":139,"seo_title":140,"seo_description":61,"update_tm":141,"read_time":142},13056703019404,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Harold’s Abstract Algebra Cheat Sheet  \n5 May 2026  \nDRAFT  \nSymbols  \n\n| Symbol | Name / Definition | Symbol | Name / Definition |\n| --- | --- | --- | --- |\n| ∅ | Empty set, set with no members | R0, R90,\u003Cbr>R180, R270 | Rotation |\n| ℕ | Natural numbers | R360/n | Cyclic Rotation |\n| ℤ | Integers (Zahlen) | H, V, D, D’ | Flip (horizontal, vertical, diagonal) |\n| ℚ | Rational numbers | 〈a〉 | The set {an | n ∈ ℤ} under •\u003Cbr>The set {na | n ∈ ℤ} under + |\n| ℝ | Real numbers | [􀜣􀜥 􀜤􀜦 ]−1 | 2x2 Matrix Inverse |\n| ℂ | Complex numbers | Zn | Group of integers modulo n |\n| F* | Nonzero Field | Zp | Zn where p a prime |\n| ⊆ | Is a subset of | mod | Modulus arithmetic |\n| ∈ | Is an element of | GL(2, F) | General Linear Group of 2x2 matrices over the field F |\n| ∞ | Infinity | gn | The group operation on g n times |\n| ° | Degrees | |G| | Order of a Group |\n| ≤, ≠, ≥ | Inequalities | |g| | Order of an Element |\n| • ∙\u003Cbr>, | Multiply | gcd (a, b) | Greatest Common Divisor |\n| ÷ | Division | lcm (a, b) | Least Common Multiple |\n| a | b | a divides b |  |  |\n| a-1 | Inverse |  |  |\n| \u003Ctab> |  |  |  |\n|  |  |  |  |\n|  |  |  |  |\n|  |  |  |  |\n|  |  |  |  |\n|  |  |  |  |\n|  |  |  |  |\n|  |  |  |  |\n|  |  |  |  |\n\nCh. 0: Preliminaries  \n\n| Definition | Description |\n| --- | --- |\n| Well Ordering Principle | Every nonempty set of positive integers contains a smallest member. |\n| Theorem 0.1:\u003Cbr>Division Algorithm | Let a and b be integers with b > 0.\u003Cbr>Then there exist unique integers q and r with the property that a = bq + r, where 0 ≤ r \u003C b.\u003Cbr>Example: For a = 17 and b = 5, the division algorithm gives 17 = 5 ⋅ 3 + 2. Here q = 3 and r = 2. |\n| Greatest Common Divisor (GCD) | 􀝃􀜿􀝀 (􀝔, 􀝕) = 􀝌􀯠􀯜􀯡1{􀰈1,􀰉1} · 􀝌􀯠􀯜􀯡2{􀰈2,􀰉2} · 􀝌􀯠􀯜􀯡􀯞{􀰈 􀳖,􀰉 􀳖} Largest positive integer that is a factor of both x and y.\u003Cbr>Think Intersection (∩) of 􀟙􀯜 , 􀟚􀯜 . |\n|  | The greatest common divisor of two nonzero integers a and b is the largest of all common divisors of a and b. We denote this integer by gcd (a, b) . |\n| Relatively Prime Integers | When gcd (a, b) = 1, we say a and b are relatively prime. |\n| Theorem 0.2:\u003Cbr>GCD Is a Linear\u003Cbr>Combination | For any nonzero integers a and b, there exist integers s and t such that gcd (a, b) = as + bt. Moreover, gcd(a, b) is the smallest positive integer of the form as + bt. |\n| Corollary | If a and b are relatively prime, then there exist integers s and t such that as + bt = 1.\u003Cbr>Example: gcd (4, 15) = 1 where 4 and 15 are relatively prime and 4 ⋅ 4 + 15(-1) = 1. |\n| Euclid’s Lemma\u003Cbr>p | ab Implies p | a or p | b | If p is a prime that divides ab, then p divides a or p divides b. |\n| Theorem 0.3:\u003Cbr>Fundamental Theorem of Arithmetic | Every integer greater than 1 is a prime or a product of primes. This product is unique, except for the order in which the factors appear.\u003Cbr>That is, if n = p1 p2 ... pr and n = q1q2 ... qs, where the p’s and q’s are primes, then r = s and, after renumbering the q’s, we have pi = qi for all i. |\n| Least Common Multiple (LCM) | 􀝈􀜿􀝉 (􀝔, 􀝕) = 􀝌􀯠􀯔􀯫1{􀰈1,􀰉1} · 􀝌􀯠􀯔􀯫2{􀰈2,􀰉2} · 􀝌􀯠􀯔􀯫􀯞{􀰈 􀳖 ,􀰉 􀳖} Smallest positive integer that is an integer multiple of both x and y.\u003Cbr>Think Union (∪) of 􀟙􀯜 , 􀟚􀯜 . |\n|  | The least common multiple of two nonzero integers a and b is the smallest positive integer that is a multiple of both a and b.\u003Cbr>We will denote this integer by lcm (a, b) .\u003Cbr>Example: lcm (4, 6) = 12 |\n| Computing ab mod n or (a + b) mod n | Let n be a fixed positive integer greater than 1. If a mod n = a’ and b mod n = b’, then\u003Cbr>(a + b) mod n = (a’ + b’) mod n\u003Cbr>(ab) mod n = (a’b’) mod n |\n\n\n| Logic Gates | A logic gate is a device that accepts as inputs two possible states (on or off) and produces one output (on or off) . This can be conveniently modeled using 0 and 1 and modulo 2 arithmetic.\u003Cbr>x AND y xy\u003Cbr>x OR y x + y + xy\u003Cbr>x XOR y x + y\u003Cbr>MAJ(x, y, z) xz + xy + yz. |\n| --- | --- |\n| Theorem 0.4:\u003Cbr>Properties of Complex Numbers | 1. Closure under addition:\u003Cbr>(a + bi) + (c + di) = (a + c) + ","cbCaicxhP7T1ojfH","https://ap.wps.com/l/cbCaicxhP7T1ojfH","pdf",2632103,26,"English","# Symbols\n# Ch. 0: Preliminaries\n## Division and gcd/lcm\n## Fundamental theorems and induction\n## Complex numbers and DeMoivre’s theorem\n## Equivalence relations","[{\"question\":\"What does the cheat sheet cover under “Symbols”?,\",\"answer\":\"It lists standard mathematical symbols and short definitions covering empty sets and number systems (ℕ, ℤ, ℚ, ℝ, ℂ), subset and element relations, modular arithmetic, inverses, inequalities, and group-related notation such as order.\"},{\"question\":\"Which core number-theory results are included in “Ch. 0: Preliminaries”?,\",\"answer\":\"It includes the Division Algorithm, statements and consequences around gcd (including a linear combination form), Euclid’s Lemma, and the Fundamental Theorem of Arithmetic, plus LCM-related definitions and examples.\"},{\"question\":\"What algebra topics beyond number theory are included?,\",\"answer\":\"It summarizes properties of complex numbers, DeMoivre’s theorem, the first and second principles of mathematical induction, and the definition of equivalence relations and their partitioning behavior.\"}]","Harold’s Abstract Algebra Cheat Sheet - Draft | PDF",1791924100,9]