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Unformatted text preview: Documentation for: MATH 323, Algebra I. Fall 2006/07. Laurence Barker, Bilkent University Source file: exams06.323.tex page 2: Final page 3: Midterm I page 4: Midterm II page 5: Makeup I page 6: Makeup II 1 MATH 323 Algebra I FINAL 10 January 2007, LJB, Bilkent University. Time allowed: 2 hours. Each question is worth 25% of the available marks. Please make sure your name is on every sheet of your script. 1: Let p be a prime. Prove that every finite group has a Sylow psubgroup. 2: Let p be an odd prime, let a be a positive integer, let q = p a and let F be a field with order q . The group GL 2 ( F ) is the group of invertible 2 × 2 matrices with entries in F . The group SL 2 ( F ) is the subgroup consisting of those matrices which have determinant 1. The group PGL 2 ( F ) is defined to be the quotient group PGL 2 ( F ) = GL 2 ( F ) /Z (GL 2 ( F )). For each of these groups, find its order, determine the Sylow psubgroups up to isomorphism, and find the number of Sylow psubgroups. 3: (a) Prove that the alternating group A 5 is simple. (b) For each n ∈ { 1 , 2 , 3 , 4 , 5 , 6 } , let G n be the stabilizer of n in A 6 . Show that the subgroups G n are isomorphic to A 5 and that they are mutually conjugate. (c) Prove that A 6 is simple by considering a normal subgroup N £ A 6 and by showing that G n ≤ N or G n ∩ N = 1 for each n . 4: In the two 4 × 4 squares below, each entry has at most four adjacent entries. For example, in both squares, 10 is adjacent to 6, 9, 11, 14. Using the theory of the symmetric and alternating groups, prove that it is impossible to get from the first square to the second square by a sequence of operations where each operation interchanges the • entry with an adjacent entry. 2 1 3 4 5 6 7 8 9 10 11 12 13 14 15 • 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 • 2 MATH 323 Algebra I MIDT ERM 1 10 November 2006, LJB, Bilkent University. Time allowed: 110 minutes. Attempt FIVE questions. Or, if you attempt all six, then the best five will be counted. Each question carries 20% of the marks....
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