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Review Sheet for MATH 411 at UMass

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This 1 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 21 views.

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Date Created: 02/06/15
Math 411 Spring 2006 Review for exam 2 Format and Coverage You will be asked to give some de nitions7 proofs7 examples7 and counter examples The exam covers sections 147 21 7 257 317 32 excluding some material that we did not discuss in class The following is a practice exam7 which is also your homework for the week do not submit Give the de nition of a coset If G is a group and H is its subgroup prove that two cosets 1H and bH either do not intersect or coincide try to prove it Without looking at the textbooknotes Let f G a G be a group homomorphism Show that the order of an element a E G is divisible by the order of its image fa E G Does there exist a homomorphism Q5 Z a Z3 gtlt Z2 that maps 2 to 17 1 Justify Let Q5 Z9 a 33 be the unique homomorphism that maps 1 to the 3 cycle 123 Determine the kernel and the image of Q5 Describe all possible onto homomorphisms from D4 to Z4 Let G be a group a Show that Q5 0 T9 TMQ 0 Q5 for any Q5 6 AutG and any 9 E G b Prove that InnG is a normal subgroup of AutG Show that V4 Z2 gtlt Z2 Bonus Prove that S4V4 Sg Hint There are three possible parti tions of the set 172374 into 2 element subsets a 172 U 3747 b 173 U 247 and c 174 U 273 Each permutation in S4 induces a permutation on the set 17 b7 c7 ie de nes an element of 83 Use the rst isomorphism theorem

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