# Cyclic Groups

A) let a be the m-cycle (123.....m). how to show that a^i is also an m-cycle if and only if i is relatively prime to m. Here a is an element of group G that generates the m-cycle.

b) How to prove that the order of an element in Sn equals the least common multiple of the lengths of the cycles in its cycle decomposition.

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#### Solution Preview

a) Proof:

We only need to show that a^i has order m. Suppose a^i has order n, then we have (a^i)^n=a^(in)=a^m=1, then m|(in), but gcd(m,i)=1 since i and m are relatively prime, then we must have m|n. On ...

#### Solution Summary

Cyclic Groups are investigated.

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