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It includes four problems totaling 100 points, covering invariants of groups under inversion, properties of conjugation powers, normality of subgroups within cyclic normal subgroups, and permutation group facts for S12 (including parity examples and order arguments). 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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 be a group and let I (G) = fx 2 G : x = x􀀀1g.  \n(a) Show that if G is abelian, then I (G) is a subgroup of G.  \n(b) Show that if G is 􀀌nite and I (G)  feg, then G must have even order.  \n(c) Give an example of a group G for which I (G) is not a subgroup of G.  \n2. Let G be a group.  \n(a) Let g; h 2 G. Prove that for all integers k 􀀕 1,  \n(h􀀀1gh)k = h􀀀1g kh:  \n(b) Let N be a normal subgroup of G and suppose that N is cyclic. Let H be a subgroup of N. Prove that H is a normal subgroup of G.  \n3. Consider the group S12 of permutations of the set f1 ; 2 ; 3 ; : : : ; 12g.  \n(a) Give an example of an even permutation 􀀛 2 S12 . Don't forget to justify your answer.  \n(b) Give an example of a permutation 􀀜 2 S12 that has order 14 . Don't forget to justify your answer.  \n(c) Prove that no element of S12 has order 13 .  \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, and suppose that xy 􀀀 yx 2 I for every x; y 2 R. Prove that the quotient ring R=I is commutative.","cbCaimH2N9SK7GXJ","https://ap.wps.com/l/cbCaimH2N9SK7GXJ","pdf",128151,"English","# Read This First\n# Questions\n## 1. Groups and inverses\n## 2. Conjugation powers and normal cyclic subgroups\n## 3. Permutations in S12","[{\"question\":\"How does the exam assign credit for problem solutions?\",\"answer\":\"To receive full credit, solution methods must be complete, logical, and understandable. You are instructed to show all work and justify answers, with final answers written in the provided spaces.\"}]","Comprehensive Examination C Algebra B Practice Exam 3 | PDF",1790754990,13]