Symmetric Group

Discussion in 'Undergraduate Math' started by khosaa, Feb 28, 2011.

  1. khosaa

    khosaa Guest

    Hi,

    I am wondering why the group of all permutations on a set of n
    elements is called a "symmetric group". (Sometimes knowing the origin
    of a word helps me to better understand it). Neither wikipedia, nor
    any of the texts I own (Hungerford's "Intro to Algera", Herstein, or
    Dummit and Foote), directly answer this question.

    I understand that the subgroup of all symmetries of a regular n-gon
    are contained within S_n, but it seems to me that not all the elements
    of S_n are in fact symmetries of a regular n-gon. (i.e. for n=4, I
    don't think the permutation (1324) is part of the octic subgroup of
    S_4).

    Thank-you,
    Fran
     
    khosaa, Feb 28, 2011
    #1
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  2. khosaa

    Chip Eastham Guest

    The symmetries of the regular n-gon (adjacency preserving) are
    known as the dihedral group of order 2n (rotations and reflections),
    and the notation D(n) or D(2n) varies from author to author.

    However your point is correct, this is not all of S_n. If memory
    serves, S_n is the symmetry group of the (n-1)-simplex, so in the
    case of S_4, look at the symmetries of the regular tetrahedron.

    regards, chip
     
    Chip Eastham, Feb 28, 2011
    #2
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  3. khosaa

    Ken Pledger Guest


    The same question came up recently with the subject "Re: etymology of
    the symmetric group", and here's what I answered on the 11th January.
     
    Ken Pledger, Feb 28, 2011
    #3
  4. khosaa

    khosaa Guest

    Hi Ken,

    I did do a search in this group under "symmetric groups" but somehow
    did not come across your response. It's kind of interesting as the
    term is actually used to *define* a type of polynomial (which as I
    understand it, are those homogeneous polynomials which do not change
    sign upon exchanging any two variable labels).

    Thanks,
    Fran
     
    khosaa, Mar 1, 2011
    #4
  5. khosaa

    scaaahu Guest

    I used one of the sentences to do search in this group and found it.
    Sometimes Google is not that smart anymore.

    I always thought symmetric group was named after symmetry.

    Anyway, thanks for the answer, I learned something today.
    http://groups.google.com/group/sci...._2+++x_3+is+invariant+under+#7ad5861f4b05e93e
     
    scaaahu, Mar 1, 2011
    #5
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