s09_mthsc851_midterm1

s09_mthsc851_midterm1 - H,N G and N C G , then (a) HN is a...

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MTHSC 851, Midterm 1 Spring 2009 Instructor: Dr. Matthew Macauley Thursday, February 19, 2009 Name Instructions Exam time is 75 minutes You may not use notes or books. PLEASE WRITE NEATLY AND COMPLETELY! Work out what you want to say on scratch paper before you write it out on the test. There are two pages of scratch paper at the end. If you have any doubt about what results you may assume, then ask . Question Points Earned Maximum Points 1 10 2 20 3 20 4 10 5 15 6 10 7 15 Total 100 Student to your left: Student to your right:
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1. (a) Exhibit an element of S 9 of order 20. (b) How many elements of order 18 are there in S 9 ?
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2. (a) Carefully define what an action of a group G on a set S is. (b) Carefully define what an orbit is, and what a stabilizer is. (c) State the orbit-stabilizer theorem.
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3. Prove all parts of the Third Isomorphism Theorem for groups: If
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Unformatted text preview: H,N G and N C G , then (a) HN is a subgroup of G ; (b) H N is a normal subgroup of H ; (c) H/ ( H N ) = HN/N . 4. Let G be a group that contains an element x G that has exactly two conjugates. Prove that G is not simple. 5. (a) Let H be a proper subgroup of a nite group G . Show that if | G | does not divide [ G : H ]! then G is not simple. (b) Show that any group of order 36 cannot be simple. (You may use the result of part (a), even if you cant prove it). 6. Let D n be the group of symmetries of a regular n-gon. If n is odd, how many 2-Sylow subgroups does D n have? 7. Suppose f : G H is a group homomorphism, with H abelian, and ker f < N < G . Show that N C G . Do not use any results about commutator subgroups....
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s09_mthsc851_midterm1 - H,N G and N C G , then (a) HN is a...

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