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# Adv Alegbra HW Solutions 57 - 57 If f G H and g G K are...

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57 If f : G H and g : G K are isomorphisms, then g 1 : K G is an isomorphism, by (ii), and so f g 1 : K H is an iso- morphism. (iii) Prove that isomorphism is an equivalence relation on any family of groups. Solution. Absent. (iv) Prove that two groups that are isomorphic to a third group are isomorphic to each other. Solution. Let f : G L and g : H L be isomorphisms. Then g 1 : L H is an isomorphism, by part (ii), and the composite g 1 f : G H is an isomorphism, by part (i). 2.68 Prove that a group G is abelian if and only if the function f : G G , given by f ( a ) = a 1 , is a homomorphism. Solution. If a , b G , where G is abelian, then ( ab ) 1 = b 1 a 1 = a 1 b 1 . Conversely, assume that ( ab ) 1 = a 1 b 1 for all a , b G . Then aba 1 b 1 = ab ( ab ) 1 = 1 , so that ab = ba . 2.69 This exercise gives some invariants of a group G . Let f : G H be an isomorphism. (i) Prove that if a G has in fi nite order, then so does f ( a ) , and if a has fi nite order n , then so does f (
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