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If $\pi$ is a cyclic permutation of a set, and $e$ an element of that set, can we say that $\pi(e)$ is the rotation of $e$ by $pi$? Or is there a better word phrase for it?

Take, for example, $\pi=(abcde)$. Then $\pi(e)=a$, and $a$ is the rotation of $e$ by $\pi$.

Pierre Carbonnelle
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  • I have heard the saying that $a$ is $\pi$ of $e$. – peterwhy Oct 11 '23 at 11:21
  • I'm not sure rotation is appropriate since there is no stabilized subspace. Typically when I want to describe a rotation I want an axis of rotation which doesn't change. I think a cyclic permutation or image are better choices. – CyclotomicField Oct 11 '23 at 11:44
  • @CyclotomicField I have no issues considering $\pi$ as a rotation. It certainly corresponds to a rotation if you interpret it as an element of the symmetry group of a pentagon (assuming the vertices are labelled the right way). – Arthur Oct 11 '23 at 11:51
  • @Arthur the rotation group of a regular pentagon stabilizes the center. I expect rotations to stabilize something rather than merely cycling them. You could add a letter to the symmetric group to ensure something is always being stabilized but it seem artificial. Also, rotations are not always periodic while cycles are. I think it's reasonable to segregate them based on these properties. – CyclotomicField Oct 11 '23 at 12:01
  • @CyclotomicField Yes, it stabilizes the center. But in a standard introductory treatment of $D_5$, the center isn't given a name. Only the five vertices. And none of those are stabilised. And this $\pi$ is exactly one of the rotations that compose half of $D_5$ (or all of $C_5$ if you don't like reflections). Insisting that the non-reflection elements of $D_n$ can't be called rotations seems extremely silly to me. – Arthur Oct 11 '23 at 12:14

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I think the easiest thing to do is to interpret $\pi$ as a function, and in that respect, $a=\pi(e)$ is called the image of $e$ (with respect to $\pi$).

Arthur
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