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Homework 07 - MATH 74 HOMEWORK 7 If you dont like your...

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MATH 74 HOMEWORK 7 “If you don’t like your analyst, see your local algebraist!” –Gert Almkvist Due Wednesday April 8th at 3:10pm. Throughout, let G be a group. Many of the following problems are from (or adapted from) Herestein’s Topics in Algebra . (1) In the following, decide and supply reasoning as to whether the given systems are groups. If they aren’t point out an axiom that fails. (a) the set of integers with a · b := a - b . (b) the set of positive integers with a · b := ab (the usual product of integers). (c) the set of all rational numbers with odd denominators, with the usual additional of rational numbers. (2) Given a, b G , prove that the equations a · x = b and y · a = b have unique solutions for x, y G . In particular, show we have cancellation laws: a · u = a · w = u = w, u · a = w · a = u = w. (3) For an element of a group, let a n denote the n -fold multiplication of a with itself. If G is a group with ( a · b ) 2 = a 2 · b 2 , prove that G is abelian.
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