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A2sol - PMath 336 Introduction to Group Theory Exercise set...

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Unformatted text preview: PMath 336: Introduction to Group Theory Exercise set 2 May 27, 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 Z : The group of integers under addition Z n : The group { ,...,n- 1 } with addition modulo n 1. (16 points) Prove that if a group G is abelian, then for any n , the set of elements of the form g n where g ∈ G is a subgroup, and so is the subset of elements that satisfy g n = e . Show the same for the subset of elements x such that, for any natural number n , x has the form g n for some g (i.e., x = g 2 1 = g 3 2 = g 4 3 = ... ) Finally, show that when G is not abelian, any of these may or may not be a subgroup (Hint: Consider the group B n ( C ) of upper-triangular matrices over the complex numbers.) Solution: If x = g n and y = h n , then xy = g n h n = ( gh ) n since G is abelian. Likewise, x- 1 = ( g n )- 1 = (...
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A2sol - PMath 336 Introduction to Group Theory Exercise set...

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