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The problems require proving basic subgroup and normal subgroup properties under group homomorphisms, analyzing finite groups with commuting involutions, determining missing images in an alternating-group permutation and computing its order, and working with ideals in rings of polynomials defined by constraints at specific real points. The exam emphasizes complete, logical, and justified solutions.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/comprehensive-examination-c-algebra-b-practice-exam-1/461969/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/comprehensive-examination-c-algebra-b-practice-exam-1/461969.png","ImageObject",300,407,{"name":92,"@type":93},"Adam","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-09-30",true,{"@type":102,"interactionType":103,"userInteractionCount":24},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What exam rules are stated before the questions begin?","Question",{"text":112,"@type":113},"The exam is closed-book and no books, notes, phones, or other aids are permitted. Cell phones must be silenced and out of sight, and answers must be written in the provided spaces with full justified work.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How is N1 defined in Question 1 and what must be proved?",{"text":117,"@type":113},"Given groups G1 and G2 and a homomorphism φ: G1 → G2, with N2 ⊆ G2, define N1 = {a ∈ G1 : φ(a) ∈ N2}. One must prove N1 is a subgroup of G1, and if N2 is normal in G2 then N1 is normal in G1.",{"name":119,"@type":110,"acceptedAnswer":120},"What does Question 4 ask about the ideal I in R[x]?",{"text":121,"@type":113},"With R the real numbers and R[x] the polynomial ring, I = {f ∈ R[x] : f(1) = f(2) = 0}. It asks to define what an ideal is, prove I is an ideal of R[x], and then show that the quotient ring R[x]/I has zero-divisors by finding nonzero classes whose product is zero.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},461969,1791543083,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":24,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":24,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":144,"preview_page_count":4},1374404737137,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Amherst College  \nDepartment of Mathematics and Statistics  \nComprehensive Examination C Algebra B Practice Exam 1  \nNumber:    \nRead This First:  \n􀀏 This is a closed-book examination. No books, notes, cell phones, electronic devices of any sort, or other aids are permitted. Cell phones are to be silenced and out of sight.  \n􀀏 Write your number (not your name) in the above space.  \n􀀏 For any given problem, you may use the back of the previous page for scratch work. Put your 􀀌nal answers in the spaces provided.  \n􀀏 Additional sheets of paper will be available if you need them. If you use an additional sheet, label it carefully and be sure to include your number.  \n􀀏 In order to receive full credit on a problem, solution methods must be complete, logical and understandable. Show all your work, and justify your answers.  \n􀀏 The Algebra Exam consists of Questions 1{4 that total to 100 points.  \nFor Department Use Only:  \nGrader \\#1:    \nGrader \\#2:    \n1. Let G 1 and G2 be groups and 􀀞 : G 1 ! G2 be a homomorphism. Suppose that N2 is a subgroup of G2 and de􀀌ne the set  \nN1 = fa 2 G 1 : 􀀞 (a) 2 N2 g:  \n(a) Prove that N1 is a subgroup of G 1 .  \n[Note: this is a standard theorem in Math 350 . Since you are being asked to prove that theorem here, you may not quote that theorem.]  \n(b) Prove that if N2 is a normal subgroup of G2 then N1 is a normal subgroup of G 1 .  \n2. Let G be a 􀀌nite group. Suppose that x and y are distinct elements of order two in G such that xy = yx. Prove that the order of G is divisible by 4 .  \n3. Suppose that 􀀛 is a permutation in the alternating group A 10 given by  \n􀀒 14 27 32 46 5 610 1 75 8 9 103􀀓 where the images of 8 and 9 have been lost.  \n(a) Determine the images of 8 and 9 under 􀀛 . Don't forget to justify your answer.  \n(b) Compute the order of 􀀛 .  \n4. Let R be a ring.  \n(a) De􀀌ne what it means for a subset I 􀀒 R to be an ideal of R.  \nIf you use any other technical terms like \\closed,\" \\subring,\" \\group,\" \\subgroup,\"etc. , you must fully de􀀌ne those terms as well.  \n(b) Let R = R [x] be the ring of polynomials with coe􀀎cients in the 􀀌eld R of real numbers. Let I 􀀒 R be the subset  \nI = ff 2 R[x] : f(1) = f(2) = 0g:  \nProve that I is an ideal of R.  \n(c) With R = R [x] and I de􀀌ned in part (b), prove that R=I has zero-divisors. That is, show that there are two nonzero elements of R=I whose product is zero in R=I.","cbCaicYoicQ5heH1","https://ap.wps.com/l/cbCaicYoicQ5heH1","pdf",124754,"English","# Instructions\n# Questions\n## Group theory\n## Alternating group permutation\n## Rings and ideals","[{\"question\":\"What exam rules are stated before the questions begin?\",\"answer\":\"The exam is closed-book and no books, notes, phones, or other aids are permitted. Cell phones must be silenced and out of sight, and answers must be written in the provided spaces with full justified work.\"},{\"question\":\"How is N1 defined in Question 1 and what must be proved?\",\"answer\":\"Given groups G1 and G2 and a homomorphism φ: G1 → G2, with N2 ⊆ G2, define N1 = {a ∈ G1 : φ(a) ∈ N2}. One must prove N1 is a subgroup of G1, and if N2 is normal in G2 then N1 is normal in G1.\"},{\"question\":\"What does Question 4 ask about the ideal I in R[x]?\",\"answer\":\"With R the real numbers and R[x] the polynomial ring, I = {f ∈ R[x] : f(1) = f(2) = 0}. It asks to define what an ideal is, prove I is an ideal of R[x], and then show that the quotient ring R[x]/I has zero-divisors by finding nonzero classes whose product is zero.\"}]","Comprehensive Examination C Algebra B Practice Exam 1 | PDF",1790762906,13]