Adv Alegbra HW Solutions 77

Adv Alegbra HW Solutions 77 - 77 (iv) Use parts (ii) and...

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77 (iv) Use parts (ii) and (iii) to give a second proof of the simplicity of A 5 . Solution. Since H contains 1, the order of H is a sum of 1 together with some of the numbers 12, 12, 15, and 20. The only such sum that divides 60 is 60 itself. 2.126 If σ,τ S 5 , where σ is a 5-cycle and τ is a transposition, prove that h σ,τ i= S 5 . Solution. Absent. 2.127 (i) For all n 3, prove that every α A n is a product of 3-cycles. Solution. We show that ( 123 ) and ( ijk ) are conjugate in A n (and thus that all 3-cycles are conjugate in A n ). If these cy- cles are not disjoint, then each f xes all the symbols outside of { 1 , 2 , 3 , i , j ]} , say, and the two 3-cycles lie in A , the group of all even permutations on these 5 symbols. Of course, A = A 5 , and, as in Lemma 2.155, ( 123 ) and ( ijk ) are conjugate in A ; a for- tiori , they are conjugate in A n . If the cycles are disjoint, then we have just seen that ( 123 ) is conjugate to ( 3 jk )
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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