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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 􀀌nite groups and 􀀞 : G 1 ! G2 be a homomorphism. Suppose that x 2 G 1 has order n 􀀕 1.  \n(a) Show that the order of 􀀞(x) divides n.  \n(b) Prove that if the order of G2 is relatively prime to n, then x is in the kernel of 􀀞 .  \n2. Let G be a group and de􀀌ne Z = fg 2 G : ga = ag for all a 2 Gg.  \n(a) Show that Z is a subgroup of G.  \n(b) Show that the subgroup Z is normal in G.  \n(c) Prove that if the quotient group G=Z is cyclic, then G is abelian.  \n3. Consider the group S9 of permutations of the set f1 ; 2 ; 3 ; : : : ; 9g. Let 􀀛; 􀀜 2 S9 be the permutations  \n􀀛 = (1 ; 4 ; 3)(9 ; 5 ; 7) and 􀀜 = (3 ; 9)(1 ; 5 ; 8) :  \n(a) Write 􀀜􀀛 2 as a product of disjoint cycles.  \n(b) Compute the order of each of 􀀛 , 􀀜, and 􀀜􀀛 2 .  \n(c) Decide whether each of 􀀛 , 􀀜, and 􀀜􀀛 2 is an even or odd permutation; don't forget to justify.  \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 I 􀀒 R be an ideal of R, S be another ring, and 􀀞 : R ! S be a ring homomorphism. Prove that if 􀀞 is surjective then 􀀞 (I) = f􀀞(x) : x 2 Ig is an ideal of S.","cbCaioChadKYSktL","https://ap.wps.com/l/cbCaioChadKYSktL","pdf",114392,"English","# Read This First\n# Questions\n## 1. Finite groups and homomorphisms\n## 2. Centralizer and normality\n## 3. Permutations: cycles, order, parity\n## 4. Ideals in rings and homomorphisms","[{\"question\":\"Is this exam open-book, and what restrictions apply?\",\"answer\":\"It is a closed-book examination. No books, notes, cell phones, electronic devices, or other aids are permitted, and cell phones must be silenced and out of sight.\"},{\"question\":\"What is the expected standard for solutions to receive full credit?\",\"answer\":\"Solution methods must be complete, logical, and understandable. You must show all your work and justify your answers for each problem.\"},{\"question\":\"What are the main subject areas covered by Questions 1–4?\",\"answer\":\"The exam covers group theory topics such as homomorphisms, orders, centralizers and normality, cyclic quotient groups implying abelian structure, and permutation parity; it also covers ring theory through the definition of ideals and the image of ideals under surjective ring homomorphisms.\"}]","Comprehensive Examination C Algebra B Practice Exam 2 | PDF",1790769898,13]