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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