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# e2sol - 1 Math 417 Exam II Solutions 1 If = 1 t is a...

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1 Math 417 Exam II Solutions: July 24, 2009 1 . If α = β 1 · · · β t is a product of disjoint r i -cycles β i , prove that α has order m = lcm { r 1 , . . . , r t } . This is Proposition 2.55(ii). 2 ( i ) . (10 points) If G is a group with Aut( G ) = { 1 } , prove that g 2 = 1 for every g G . If G is not abelian, then there is a G which in not in the center Z ( G ); that is, there is some g G with aga - 1 = g . Hence, γ a : G G , conjugation by a , defined by x axa - 1 , is an automorphism of G that is not the identity 1 G because aga - 1 = g . If G is abelian, then we have proved that ι : g g - 1 is an automorphism of G . Now ι = 1 G if there exists x G with x = x - 1 . Hence, ι = 1 G if g = g - 1 for all g G ; that is, if g 2 = 1 for all g G . Remark: Here is a sketch of how to treat this last case.The hypothesis g 2 = 1 for all g G implies that G can be viewed as a vector space with scalars in I 2 . If | G | > 2, then G has a basis x, y, . . . . Define an automorphism ϕ of G by ϕ ( x ) = y , ϕ ( y ) = x , and ϕ ( z ) = z for all other basis elements, if any. Hence, if Aut( G ) = { 1 } , then | G | ≤ 2.

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e2sol - 1 Math 417 Exam II Solutions 1 If = 1 t is a...

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