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### prelim2

Course: MATH 432, Fall 2008
School: Cornell
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Word Count: 265

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432 Math Prelim 2 Due date Monday April 16, 2007 Problem 1. (2.88 from the textbook) Let G be a finite group generated by two elements of order 2. Prove that G is isomorphic to the dihedral group D2n for some n. Problem 2. Let G be the symmetric group Sym(n) for n = 4, 5 or 7. Show that any automorphism of G is inner, i.e. there exists h G such that (g) = hgh-1 for any g G. Hint: Show that preserves the set of...

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432 Math Prelim 2 Due date Monday April 16, 2007 Problem 1. (2.88 from the textbook) Let G be a finite group generated by two elements of order 2. Prove that G is isomorphic to the dihedral group D2n for some n. Problem 2. Let G be the symmetric group Sym(n) for n = 4, 5 or 7. Show that any automorphism of G is inner, i.e. there exists h G such that (g) = hgh-1 for any g G. Hint: Show that preserves the set of all transpositions. Problem 3. (2.125 from the textbook) Show that the alternating group Alt(5) has exactly 5 conjugacy classes and determine their sizes. Use this information to show that Alt(5) is simple. Problem 4. Let f (x) be a polynomial with coefficients in the finite field Fp . Use the inclusion Fp Matn (Fp ) to an define "evaluation map" fn : Matn (Fp ) Matn (Fp ). a) Let Func Matn (Fp ), Matn (Fp ) denote the ring of functions from Matn (Fp ) to Matn (Fp ) with pointwise operations. Prove that map n : Fp [x] Func Matn (Fp ), Matn (Fp ) defined as n (f ) = fn is a ring homomorphism. b) Show that ker n is a non-zero ideal in Fp [x] for all n. c) Prove that ker n = (0). Equivalently, show that if f (x) is not zero then there exists n su...

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University of Florida - COT - 3502
University of Florida - COT - 3502
University of Florida - COT - 3502
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University of Florida - COT - 3502
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