MATH 74 HOMEWORK 7“If you don’t like your analyst, see your local algebraist!”–Gert AlmkvistDue Wednesday April 8th at 3:10pm. Throughout, letGbe a group. Many of the following problems arefrom (or adapted from) Herestein’sTopics in Algebra.(1) In the following, decide and supply reasoning as to whether the given systems are groups. If theyaren’t point out an axiom that fails.(a) the set of integers witha·b:=a-b.(b) the set of positive integers witha·b:=ab(the usual product of integers).(c) the set of all rational numbers with odd denominators, with the usual additional of rationalnumbers.(2) Givena, b∈G, prove that the equationsa·x=bandy·a=bhave unique solutions forx, 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, letandenote then-fold multiplication ofawith itself. IfGis a groupwith (a·b)2=a2·b2, prove thatGis abelian.
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