[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-573822-105":53,"doc-detail-573822-en":126},{"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":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","kayla-orlinsky-algebra-exam-cheat-sheet","Kayla Orlinsky Algebra Exam Cheat Sheet","","Kayla Orlinsky Algebra Exam Cheat Sheet provides a compact set of group theory and related algebra results for exam preparation. It summarizes key theorems and lemmas on isomorphisms, Sylow theorems and their numeric constraints, recognition of direct and semidirect products, and classification of finitely generated abelian groups. Additional sections cover solvable groups, properties of symmetric groups, automorphism group formulas, group actions, and worked examples.",{"@graph":63,"@context":118},[64,80,101],{"@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/kayla-orlinsky-algebra-exam-cheat-sheet/573822/",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/kayla-orlinsky-algebra-exam-cheat-sheet/573822.png","ImageObject",442,249,{"name":88,"@type":89},"Terk","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-16","2026-10-14",true,{"@type":98,"interactionType":99,"userInteractionCount":79},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What do the Sylow theorems say about the number of Sylow p-subgroups?","Question",{"text":108,"@type":109},"They guarantee Sylow p-subgroups exist for each prime p dividing |G|, and for fixed p the Sylow p-subgroups are conjugates. The number np satisfies np ≡ 1 (mod p), divides m when |G| = pnm with gcd(p, m) = 1, and equals [G : NG(P)].","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How can you recognize a direct product using subgroups H and K?",{"text":113,"@type":109},"A direct product criterion states that G is isomorphic to H × K if and only if G has two normal subgroups H and K, HK = G, and H ∩ K = {e}.",{"name":115,"@type":106,"acceptedAnswer":116},"What is a group action and how is it represented as a homomorphism?",{"text":117,"@type":109},"A group action of G on a set X defines a homomorphism ϕ: G → S|X| via ϕ(g) = σg, where σg maps x to g · x. Two useful exam actions highlighted are conjugation on Sylow p-subgroups and left multiplication on cosets.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},573822,1792012938,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":79,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":35,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":140},1099525198933,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Kayla Orlinsky Algebra Exam Cheat Sheet  \nThis color corresponds to Group and Field Theory  \nThis color corresponds to Ring and Module Theory  \n􀀍á1Group Classification Theory 1á􀀍  \nTheorem 1 . Isomorphism Theorems  \nG􀀞ker(ϕ)  Im (ϕ) H􀀞N ∩ H  NH􀀞N (G/K)􀀞(H/K)  G􀀞H  \nTheorem 2 . Sylow Theorems If: |G| \u003C ∞  \nThen:  \n(1) Sylow p-subgroups exist for all p  \n(2) For fixed p, Sylow p-subgroups are conjugates  \n(3) The number of Sylow p-subgroups np satisfies the following:  \ná np ≡ 1 mod p  \ná If G = pnm where gcd (p, m) = 1, then np divides m  \ná np = [G : NG (P )]  \nTheorem 3 . Recognizing Direct Products  \nG  H × K ⇐⇒  \ná  \ná G has two normal subgroups H, K á HK = G  \ná H ∩ K = {e}  \nTheorem 4 . Recognizing Semi-Direct Products If:  \ná G has a subgroup H and a normal subgroups N á HN = G  \ná H ∩ N = {e}  \nThen: G  N oϕ H assuming there exists a non-trivial homomorphism ϕ : H → Aut (N) .  \n***Note that if a semi-direct product exists, then its multiplication is given by nhn−1 =ϕ (h)(n) for h ∈ H , n ∈ N.  \nTheorem 5 . Isomoprhic Semi-Direct Products Given N oϕ1 H and N oϕ2 H with ϕ 1 , ϕ2 : H → Aut(N)  \nIf:  \ná there exists an automorphism σ : H → H such that ϕ 1 ◦ σ = ϕ2  \ná OR there exists an automorphism α : N → N so  \nϕ 1 (h) = α ◦ ϕ2 (h) ◦ α −1 for all h ∈ H  \ná OR a there exists both σ and α so (ϕ1 ◦ σ)(h) = α ◦ ϕ2 (h) ◦ α −1 for all h ∈ H  \nThen:  \nN oϕ1 H  N oϕ2 H  \nExample 1 .  \nDetermine all semi-direct products up to isomorphism of Z 15 o Z67  \nFirst, let Z3  hai, Z5  hbi, and Z67  hci.  \nThen since Aut(Z67 )  Z66 we have that ϕ (b) =id since 5 does not divide the order of Z66 and ϕ (a) = α where α has order 3.  \nSince Z66 is abelian, there are exactly two non-trivial options for α and one will be the square of the other. Namely, if ϕ 1 (a) = α and ϕ2 (a) = α2 , then ϕ 1 (a2 ) = ϕ2 (a) and since a →7 a2 is an automorphism of Z3 , these will generate isomorphic semi-direct products.  \nOne can check that α3 (c) = α2 (c29 ) = α(c37 ) = c has order 3 and defines multiplication for G given by bcb−1 = ϕ(b)(c) = c and aca−1 = ϕ(a)(c) = c29 .  \nThus, Z 15 o Z67  ha, b, c | a3 = b5 = c67 = 1, ab = ba, bc = cb, ac = c29 ai.  \nTheorem 6 . Classification of Finitely Generated Abelian Groups If: G is a finitely generated abelian group  \nThen:  \nG  Zm ⊕ Zn1 ⊕ Zn2 ⊕ · · · ⊕ Znm ni |ni+1∀i.  \n***Note that it is possible to break each of the Zni into its prime power divisors andreorder, however, the primes may not be distinct.  \nFor example, Z 12 ×Z2 = Z2 ×Z22 ×Z3 which is of course different from Z2 ×Z2 ×Z2 ×Z3 .   \nDefinition 1 . Solvable Groups  \nA group G is solvable if there exists a subnormal series  \n{e} E Gn E Gn−1 E · · · E G0 = G Gi−1/Gi abelain ∀i  \nLemma 1 . Facts about Solvable Groups  \n1 Subgroups and quotients of solvable groups are solvable  \n1 If N is normal in G and solvable, and G/N is solvable, then G is solvable  \n1 Sn is not solvable for n ≥ 5 (S3 and S4 are solvable)  \nLemma 2 . Useful Results that Should be Reproved For |G| \u003C ∞  \n1 If P is a Sylow p-subgroup of a normal subgroup N E G and P E N, then P is normal in G.  \n1 If p is the smallest prime dividing |G| , then any subgroup of index p is normal in G.  \nLemma 3 . Crucial (and Citeable) Results For |G| \u003C ∞  \n1 The product of a subgroup and a normal subgroup is again a subgroup  \n1 If |HK| = ||~~ ~~HHKK|| = |G| then HK = G even if neither H nor K is normal  \n1 From the class equation: p-groups (groups of order pn for p prime) have non-trivial centers.  \n1 Inductively on the previous result: p-groups are solvable  \n1 Groups of order p2 are abelian  \n1 Groups of order pq where p does not divide q − 1 are abelian  \n1 If all of the Sylow subgroups of G are normal, then G is a direct product of its Sylow subgroups.  \nLemma 4 . Facts about the Symmetric Group In Sn:  \n1 Any cycle σ can be written as a product of transpositions: an even number of transpositions means σ is even, an odd number of transpositions means σ is odd  \n1 A k-cycle is even when k is odd, and odd when k","cbCaiknVxx9eWrvc","https://ap.wps.com/l/cbCaiknVxx9eWrvc","pdf",288612,"English","# Group Classification Theory\n## Isomorphism Theorems\n## Sylow Theorems\n## Direct and Semi-Direct Products\n## Isomorphic Semi-Direct Products\n# Abelian and Solvable Groups\n## Classification of Finitely Generated Abelian Groups\n## Solvable Groups\n# Symmetric Groups and Automorphisms\n## Symmetric Group Facts\n## Automorphism Groups\n# Group Action\n## Definition and Useful Actions\n# Examples\n## Semi-Direct Products Example\n## No Simple Groups of Order 600","[{\"question\":\"What do the Sylow theorems say about the number of Sylow p-subgroups?\",\"answer\":\"They guarantee Sylow p-subgroups exist for each prime p dividing |G|, and for fixed p the Sylow p-subgroups are conjugates. The number np satisfies np ≡ 1 (mod p), divides m when |G| = pnm with gcd(p, m) = 1, and equals [G : NG(P)].\"},{\"question\":\"How can you recognize a direct product using subgroups H and K?\",\"answer\":\"A direct product criterion states that G is isomorphic to H × K if and only if G has two normal subgroups H and K, HK = G, and H ∩ K = {e}.\"},{\"question\":\"What is a group action and how is it represented as a homomorphism?\",\"answer\":\"A group action of G on a set X defines a homomorphism ϕ: G → S|X| via ϕ(g) = σg, where σg maps x to g · x. Two useful exam actions highlighted are conjugation on Sylow p-subgroups and left multiplication on cosets.\"}]","Kayla Orlinsky Algebra Exam Cheat Sheet | PDF",1791947484,6]