Unformatted text preview: S 4 which commute with (1 2)(3 4). Let n ∈ Z + and let g ∈ A n . Suppose 4n and  g  ≤ 2. Considering A n as acting on { 1 , . . . , n } , prove that gi = i for some i where 1 ≤ i ≤ n . Since the order of g is 1 or 2, when we write it as a product of disjoint cycles, only 2cycles can occur (and 1cycles, which may be omitted). Suppose g has no ﬁxed points (i.e. no such i exists). Then g must be a product of n / 2 disjoint 2cycles. Since n / 2 must be an integer and 4n , we deduce that g is a product of an odd number of transpositions. This contradicts the fact that g ∈ A n ....
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This note was uploaded on 01/02/2012 for the course MATH 5125 taught by Professor Palinnell during the Fall '07 term at Virginia Tech.
 Fall '07
 PALinnell
 Math, Algebra

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