Math 561 HFall 2011Homework 1Drew Armstrong1.LetGbe a group. Prove that the identity element ofGis unique (that is, there is onlyone element ofGsatisfying the defining property of an identity element). This justifies ouruse of the special symbol “e”.2.LetGbe a finite group.(a) Show that there are an odd number ofx∈Gsuch thatx3=e.(b) Show that there are an even number ofx∈Gsuch thatx26=e.(Hint: What is the inverse ofxn?)3.LetGbe a finite group.For alla, b∈G, show thatabandbahave the same order aselements ofG.4.LetGbe a group and fix an elementg∈G. Define a functionφg:G→Gbyφg(h) =ghg-1for allh∈G.(a) Prove thatφgis a bijection (one-to-one and onto).(b) Prove thatφg(ab) =φg(a)φg(b) for alla, b∈G.These two properties mean thatφgis anautomorphism(a “symmetry”) ofG.5.Suppose that 1,9,16,22,53,74,79,81 are eight members of a nine-element subgroup of(Z/91Z)×. Which element has been left out? Recall: (Z/91Z)×
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