Permutation is Cyclic iff At Most One Non-Trivial Orbit
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Theorem
Let $S$ be a set.
Let $\rho: S \to S$ be a permutation on $S$.
Then:
- $\rho$ is a cyclic permutation
Proof
Necessary Condition
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Sufficient Condition
Recall the definition of cyclic permutation.
Let:
- $\rho = \begin {bmatrix} i & \map \rho i & \ldots & \map {\rho^{k - 1} } i \end{bmatrix}$
Note that the orbit $\set {i, \map \rho i, \ldots, \map {\rho^{k - 1} } i}$ is the only non-trivial orbit.
(If $k = 1$, then the above-mentioned orbit is also trivial.)
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$\Box$
Also see
Some sources use this result as a definition for a cyclic permutation.
Sources
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.6$