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Unformatted text preview: and the result follows. Prove that  Aut ( Q 8 )  is a multiple of 4. We shall use the usual notation for the elements of Q 8 , that is { 1 , i , j , k } . Then Z ( Q 8 ) = { 1 } , so in particular  Z ( Q 8 )  = 2. Therefore  Q 8 / Z ( Q 8 )  = 4 and we deduce that  Inn ( Q 8 )  = 4. Since Inn ( Q 8 ) Aut ( Q 8 ) , the result now follows from Lagranges theorem. Let G be a group of order 200. Prove that there exists H G such that 1 6 = H 6 = G . 200 = 2 3 * 5 2 . Therefore the number of Sylow 5subgroups divides 8 and is congruent to 1 modulo 5. Therefore G has exactly one Sylow 5subgroup, and so the Sylow 5subgroup (which has order 25, so cannot be 1 or G ) is normal....
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 Spring '08
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 Math, Algebra

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