Answer to Question #108183 in Abstract Algebra for Garima Ahlawat

Question #108183
Let a = (3 4 5 2 1) and b = (5 3 2 4 1) in S7. Write ab as a product of disjoint permutations. Further, is ab even? Why or why not?
1
Expert's answer
2020-04-08T12:32:50-0400

Given that ,

"a=" (3 4 5 2 1) and "b=" ( 5 3 2 4 1 ) are elements of "S_7" .

"ab=" ( 3 4 5 2 1 )( 5 3 2 4 1 )"=" (1 2 5 4 3 )

Again, "ab=" ( 1 2 5 4 3 )=( 1 3 )( 1 4 )(1 5 )( 1 2 )

Because we know that every permutation in "S_n,\\ n>1," is a product of 2-cycles. However the decompositions of a permutation into a product of 2-cycles are not unique but the number of 2-cycles is fixed, i.e., if "\\alpha,\\beta\\in S_n \\ and \\ \\alpha=\\alpha_1.\\alpha_2..\u2026...\\alpha_r" and

"\\beta=\\beta_1.\\beta_2...\u2026.\\beta_s" , where "\\alpha's \\ and \\ \\beta's" are 2-cycles . Then "s=r".


But we know that a permutation that can be expressed as a product of an even number of 2-cycles is called an even permutation.

Hence by definition, "ab" is an even permutation.


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