Solving Permutation Group Homework: Part (f) Explained

[roam]

Homework Statement

I have problems understanding part (f) of the following worked example:

[PLAIN]http://img7.imageshack.us/img7/5557/61793282.gif

The Attempt at a Solution

So in part (f), when calculating (\sigma \tau)^{9000}, how does (\sigma \tau)^{818 \times 11} (\sigma \tau)^2 reduce to (\sigma \tau)^2? What happens to the (\sigma \tau)^{818 \times 11} part?

Similarly in (\sigma \tau)^{-21}=(\sigma \tau)^{-2 \times 11} (\sigma \tau)^1 = (\sigma \tau)^1

why has the "(\sigma \tau)^{-2 \times 11}" been omitted?

 

[Office_Shredder]

What is (\sigma \tau)^11 equal to? Hint: You know that it has order 11

 

[roam]

What is (\sigma \tau)^11 equal to? Hint: You know that it has order 11

Well, (\sigma \tau)^{11}=e where e is the identity. But what happens to the (\sigma \tau)^{-2}? Why does the "-2" disappear (since anything multiplied by the identity is itself)?

 

[Office_Shredder]

(\sigma \tau)^{-2 \times 11} = ((\sigma \tau)^{11})^{-2}