I am wondering why the terminology for cyclic groups is not used for permutations.
If we have a permutation $\sigma$ on a set $A$, then for any element $a$ of $A$ we can define the integer power $a^n$ recursively in the following way:
- $a^0 = a$,
- $a^{n+1} = \sigma(a^n)$,
- $a^{n-1} = \sigma^{-1}(a^n)$,
where $\sigma(x)$ is the image element of $x$, and $\sigma^{-1}(x)$ is the element which image is $x$.
Thus, every element $a$ of $A$ generates the subset of powers of $a$ (let's denote it $[a]$).
We can say that $[a]$ is a "subpermutation" of $\sigma$ meaning $\sigma$ defines a bijection on $[a]$.
Clearly, "subpermutation" means there is a decomposition of $\sigma$ onto two disjoint permutations:
- $\sigma = \sigma([a])\ \circ\ \sigma(\overline{[a]})$,
where $\sigma([a])$ is a permutation of $[a]$, and $\sigma(\overline{[a]})$ is a disjoint permutation of the complementary subset (disjoint permutations commutes).
The "subpermutation" $[a]$ has exactly the same relation to $\sigma$ as cyclic subgroup to group, so we can apply all the definitions and statements for cyclic (sub)groups to permutations.
In particular, we must agree that a cyclic permutation can be finite or infinite.
However, all the sources I've seen insist that a cyclic permutation is finite:
Definition of cyclic permutation
https://en.wikipedia.org/wiki/Cyclic_permutation
https://mathworld.wolfram.com/CyclicPermutation.html
Is there a flaw in the definition of the integer power of an element of a permutation?