1. ## symmetric groups 2

Evaluate all the pwers of each permutation s (ie find s^k for all k).

a) 1 2 3 4 5 6
2 3 4 5 6 1

b) 1 2 3 4 5 6 7
2 1 3 4 6 5 7

c) 1 2 3 4 5 6
6 4 5 2 1 3

2. Originally Posted by mpryal
Evaluate all the pwers of each permutation s (ie find s^k for all k).

a) 1 2 3 4 5 6
2 3 4 5 6 1
I do the first one partially and you try the other ones.
Let $\displaystyle s$ the permutation $\displaystyle \begin{bmatrix} 1&2&3&4&5&6\\2&3&4&5&6&1\end{bmatrix}$.

We have:
$\displaystyle s^2(1) = s(s(1)) = s(2) = 3$
$\displaystyle s^2(2) = s(s(2)) = s(3) = 4$
$\displaystyle s^2(3) = s(s(3)) = s(4) = 5$
$\displaystyle s^2(4) = s(s(4)) = s(5) = 6$
$\displaystyle s^2(5) = s(s(5)) = s(6) = 1$
$\displaystyle s^2(6) = s(s(6)) = s(1) = 2$

Thus, $\displaystyle s^2$ is the permutation $\displaystyle \begin{bmatrix} 1&2&3&4&5&6\\ 3&4&5&6&1&2 \end{bmatrix}$

We have:
$\displaystyle s^3(1) = s(s^2(1)) = s(3) = 4$
$\displaystyle s^3(2) = s(s^2(2)) = s(4) = 5$
$\displaystyle s^4(3) ...$
...

Thus, $\displaystyle s^3$ is the permutation $\displaystyle \begin{bmatrix} 1&2&3&4&5&6\\ 4&5&..&..&..&.. \end{bmatrix}$

Fill in the ".." 's.