# A4sol - PMath 336 Introduction to Group Theory Exercise set...

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Unformatted text preview: PMath 336: Introduction to Group Theory Exercise set 4 June 11, 2008 Solution should be submitted by the end of the Monday lecture of the following week, either in class or in the submission box. You may not submit joint or identical works. Notation GL n ( S ) : Invertible n × n matrices with coefficients in S , under multiplication D n : The group of symmetries of the regular n-gon S n : The group of permutations of 1 ,...,n Z : The group of integers under addition Z n : The group { ,...,n- 1 } with addition modulo n U n : The group of numbers less than n and prime to n , with multiplication mod n . Questions 1. [8] Prove that the map g 7→ g- 1 is an isomorphism if and only if G is abelian. Use this to prove that a group in which all elements have order at most 2 is abelian. Solution: In any group, ( xy )- 1 = y- 1 x- 1 . To be an isomorphism, we need ( xy )- 1 = x- 1 y- 1 , hence we need x- 1 y- 1 = y- 1 x- 1 for all x,y , which precisely means that G is abelian. If any element has order at most 2, then g 7→ g- 1 is the identity function, which is always an isomorphism.is the identity function, which is always an isomorphism....
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A4sol - PMath 336 Introduction to Group Theory Exercise set...

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