Answer to Question #109071 in Abstract Algebra for Durgesh singh

Question

b) Let 







σ =
3 4 5 2 1
1 2 3 4 5
and 







τ =
5 3 2 4 1
1 2 3 4 5
in . S7
Write στ as a product
of disjoint permutations.
Further, is στ even? Why, or why not?

Answer 1

Given that ,

$\sigma=\begin{bmatrix} 3&4&5&2&1 \\ 1&2&3&4&5 \end{bmatrix},\gamma =\begin{bmatrix} 5&3&2&4&1 \\ 1&2&3&4&5 \end{bmatrix} \in S_7$

$\sigma\gamma=\begin{bmatrix} 3&4&5&2&1 \\ 1&2&3&4&5 \end{bmatrix}\begin{bmatrix} 5&3&2&4&1\\ 1&2&3&4&5 \end{bmatrix}$

$=\begin{bmatrix} 1&2&3&4&5 \\ 3&1&4&2&5 \end{bmatrix}$ $=(1 \ 3 \ 4 \ 2)$ .

$=(1 \ 2)(1 \ 4)(1 \ 3)$

Since we know that every permutation in $S_n,n>1,$ is a product of 2-cycles. However, the decomposition of a permutation into a product of two cycles is not unique, but the number of 2-cycles is equal in every decomposition,

i.e., if$\, \alpha=\beta_1.\beta_2.......\beta_r \ \ and \ \alpha=\gamma_2.\gamma_2............\gamma_s$

where $\beta's \ and \ the \ \gamma's$ are 2-cycles.

Then $r=s$ .

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

Hence by definition, $\sigma\gamma$ is an odd permutation.