4107_midterm2_2009

4107_midterm2_2009 - Math 4107, Midterm 2, Fall 2009...

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Unformatted text preview: Math 4107, Midterm 2, Fall 2009 November 12, 2009 1. Define the following terms. a. Ring (list the axioms). b. Centralizer of an element g in a group G. c. State the three Sylow theorems. d. Cayley’s Theorem. e. Alternating group. 2. Express the following permutation in disjoint cycle form (composition here works from right-to-left): (1 7 5 6 3)(2 1 4)(4 7 5 2 3) ∈ S7 . 3. Determine the center Z of Sn for n ≥ 2, and prove your answer. 4. Suppose G is a group of order pq , p < q both prime, such that G acts non-trivially on a set X having size q . Prove that G is abelian. 5. Determine the number of abelian groups of order 34 · 52 , and list one example from each isomorphism class. 1 ...
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