Question

Exercitiul 1 punctul c), va rog !! ​

Answer 1

Explicație pas cu pas:

Cautam cel mai mic k nenul astfel incat sa avem: $\sigma_2^k=e$.

Calculam: $\sigma_2^2=\sigma_2*\sigma_2$.

$\sigma_2^2=\left(\begin{array}{ccccc}1&2&3&4&5\\2&1&4&5&3\end{array}\right) *\left(\begin{array}{ccccc}1&2&3&4&5\\2&1&4&5&3\end{array}\right) =\left(\begin{array}{ccccc}1&2&3&4&5\\1&2&5&3&4\end{array}\right)$

Calculam: $\sigma_2^3=\sigma_2*\sigma_2^2$.

$\sigma_2^3=\left(\begin{array}{ccccc}1&2&3&4&5\\2&1&4&5&3\end{array}\right) *\left(\begin{array}{ccccc}1&2&3&4&5\\1&2&5&3&4\end{array}\right) =\left(\begin{array}{ccccc}1&2&3&4&5\\2&1&3&4&5\end{array}\right)$

Calculam: $\sigma_2^4=\sigma_2*\sigma_2^3$.

$\sigma_2^4=\left(\begin{array}{ccccc}1&2&3&4&5\\2&1&4&5&3\end{array}\right) *\left(\begin{array}{ccccc}1&2&3&4&5\\2&1&3&4&5\end{array}\right) =\left(\begin{array}{ccccc}1&2&3&4&5\\1&2&4&5&3\end{array}\right)$

Calculam: $\sigma_2^5=\sigma_2*\sigma_2^4$.

$\sigma_2^5=\left(\begin{array}{ccccc}1&2&3&4&5\\2&1&4&5&3\end{array}\right) *\left(\begin{array}{ccccc}1&2&3&4&5\\1&2&4&5&3\end{array}\right) =\left(\begin{array}{ccccc}1&2&3&4&5\\2&1&5&3&4\end{array}\right)$

Calculam: $\sigma_2^6=\sigma_2*\sigma_2^5$.

$\sigma_2^6=\left(\begin{array}{ccccc}1&2&3&4&5\\2&1&4&5&3\end{array}\right) *\left(\begin{array}{ccccc}1&2&3&4&5\\2&1&5&3&4\end{array}\right) =\left(\begin{array}{ccccc}1&2&3&4&5\\1&2&3&4&5\end{array}\right) =\e$

Cum cel mai mic k care satisface relatia data este 6 si observand ca puterile lui $\sigma$ se repeta din 6 in 6, inseamna ca $k\in M_6$ si, deci, 6 | k.