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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

1. Some minimal content, and a couple of examples.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeDec 31st 2018

“Hence these are the subgroups of symmetric groups.”

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeDec 31st 2018

oh, sorry, now I see that this is stated later as the definition. Hm, maybe it’s still worthwhile to say it right away in the Idea-section, too.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeDec 31st 2018

Do rotation permutations form a group, as it seems to suggest here?

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeDec 31st 2018

Re rotation permutation: yes, they form a group because it’s just transporting the structure of the group $\mathbb{Z}_n \hookrightarrow Aut(\{1, \ldots, n\})$ across the given bijection $i$ on which the definition depends. But the language “rotation permutation of $X$” could easily invite confusion since it suppresses mention of this $i$ which is actually crucial. Clearly rotations relative to different $i$’s do not compose.

At first the article had the title cyclic permutation which could be defined as a transitive $\mathbb{Z}$-set structure on $X$: $\sigma: \mathbb{Z} \to Aut(X)$ (normally considered given as $\sigma(1)$; also this definition allows the case where $X$ to be countably infinite, although usually one would constrain to $X$ finite). That would be the notion of rotation permutation but untethered to a particular $i$.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeDec 31st 2018
• (edited Dec 31st 2018)

Yes, the cyclic group.

[edit: ah, overlapped with Todd]

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeDec 31st 2018

Isn’t the wording misleading at rotation permutation?

A rotation permutation is, roughly speaking, a permutation in which, if we view the elements of a finite set as people standing in a circle, everybody shifts one step to the right, or everybody shifts one step to the left.

This sounds to me like a generator.

2. Thanks David, fixed now.

3. Also attempted to improve the rotation permutations example according to Todd’s remarks (thanks!).