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]

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