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Definition

For a finite set $X$, a cyclic permutation on $X$ is a permutation $\sigma \colon X \to X$ such that the induced group homomorphism $\mathbb{Z} \to Aut(X)$ from the integers to the automorphism group (i.e. the symmetric group) of $X$, sending $n \in \mathbb{Z}$ to $\sigma^n$, defines a transitive action.

One may visualize the elements of $X$ as points arranged on a circle spaced equally apart, with $\sigma(x)$ the next-door neighbor of $x$ in the counterclockwise direction, hence the name.

See also rotation permutation.

Related concepts

• cyclic group