Given permutation are,
"a=\\begin{bmatrix}\n 1 & 2 & 3 & 4&5 \\\\\n 3&4&5&2&1\n\\end{bmatrix}" and "b=\\begin{bmatrix}\n 1&2&3&4&5 \\\\\n 5&3&2&4&1\n\\end{bmatrix}"
Now ,"ab=\\begin{bmatrix}\n 1&2&3&4&5 \\\\\n 3&4&5&2&1\n\\end{bmatrix}\\begin{bmatrix}\n 1&2&3&4&5 \\\\\n 5&3&2&4&1\n\\end{bmatrix}"
"=\\begin{bmatrix}\n 1&2&3&4&5 \\\\\n 1&5&4&2&3\n\\end{bmatrix}" "=(2 \\ 5 \\ 3 \\ 4 \\ )"
"=(2 \\ 4)(2 \\ 3)(2 \\ 5)"
Since ,Every permutation in "S_n ,n>1," is a product of 2-cycles.
However this is not the only way a permutation can written as a product of 2-cycles but the number of 2-cycles are always equal
i,e if "\\alpha \\in S_n" and "\\alpha=\\beta_1 \\beta_2.......\\beta_r \\ \\text{and} \\ \\alpha=\\gamma_1\\gamma_2.......\\gamma_s"
Where "\\beta's \\ \\text{and} \\ \\gamma's" are 2-cycles.
Then "r=s."
Again we known that , A permutation that can be expressed as a product of an odd number of 2-cycles is called an odd permutation.
Hence, "ab" is an odd permutation.
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