Exam2 - G = R a b = | ab |[10 pts(b G = Z a b = a b ab[10...

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MATH 373: Exam 2 Thursday, April 1, 2010 Answer all questions in the answer booklet. Please do these in order. You are expected to justify your answers in a manner that an average MATH 373 student should be able to follow. This includes sentences for clarification. Point values are on the right hand side. They (should) sum up to 100. 1. True or False. Justify your answers. (a) If G is a simple group of order p n , where p is prime, then n = 1. [5 pts] (b) If G is a group and K C G then every homomorphism G G/K has kernel K . [5 pts] 2. Determine if the following sets G with the operation indicated form a group. If G forms a group, prove that it satisfies all of the axioms. If not, pount out which group axioms fail. (a)
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Unformatted text preview: G = R- { } , a * b = | ab | [10 pts] (b) G = Z , a * b = a + b + ab [10 pts] 3. Find the right cosets of the subgroup H = h (1 , 1) i in Z 2 × Z 4 . [10 pts] 4. Let G be an abelian group (possibly infinite) and let the set T = { a ∈ G | a m = e for some m > 1 } . (a) Prove T is a subgroup of G . [10 pts] (b) Prove that G/T has no element - other than its identity element - of finite order. [15 pts] 5. Suppose | G | = 10 and G has a subgroup of order 2 which is not normal. Prove that G is isomorphic to a subgroup of S 5 . [15 pts] 6. Determine all groups (up to isomorphism) of order 4225 = 5 2 · 13 2 . Justify your answer completely. [20 pts]...
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