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Permutations and R-Cycles

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1. If alpha is an r-cycle, show that alpha^r = (1). [There's a hint that
If alpha = (i sub 0 ... i sub r-1), show that alpha ^k(i sub 0) = i sub k.]

2. Show that an r-cycle is an even permutation if and only if r is odd.

3. If alpha is an r-cycle and 1<k<r, is alpha^k an r-cycle?

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Solution Summary

Permutations and R-Cycles are investigated.

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Proof:
1. Since alpha is an r-cycle, the alpha=(i_1,i_2,...,i_r), where i_k means
i sub k. The cycle (i_1,i_2,...,i_r) means that alpha(i_1)=i_2, alpha(i_2)=i_3,
..., alpha(i_k)=i_(k+1), ..., alpha(i_r)=i_1.
For example, alpha=(123), the alpha(1)=2, alpha(2)=3, alpha(3)=1.
We consider alpha^k (alpha to ...

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