Answer to Question #108843 in Abstract Algebra for Garima Ahlawat

Question #108843
Let a = (1 2 3 4 5
3 4 5 2 1) and b =(1 2 3 4 5
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
2020-04-10T12:08:03-0400

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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