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nLab > Latest Changes: permutation group

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

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 31st 2018

    added the line:

    “Hence these are the subgroups of symmetric groups.”

    diff, v2, current

    • 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.

    diff, v2, current

    • 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 ℤ n↪Aut({1,…,n})\mathbb{Z}_n \hookrightarrow Aut(\{1, \ldots, n\}) across the given bijection ii on which the definition depends. But the language “rotation permutation of XX” could easily invite confusion since it suppresses mention of this ii which is actually crucial. Clearly rotations relative to different ii’s do not compose.

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

    • 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.

  8. Thanks David, fixed now.

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

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