- #1
Mr Davis 97
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- 44
Let ##\sigma \in S_n##, and ##|\sigma| = r##. Then, we'll assume that ##\sigma## can be decomposed into a product of disjoint cycles, which commute with each other. So ##\sigma = c_1c_2 \dots c_k##. Then ##\sigma^r = (c_1c_2 \dots c_k)^r## = 1. Since the cycles commute, we have ##c_1^rc_2^r \dots c_k^r = 1##. Now, I am a little stumped. How can I conclude that ##c_1^r = c_2^r = \cdots = c_k^r = 1##?