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Math Help - Symmetric Group

  1. #1
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    Symmetric Group

    For every nonempty set A, the algebraic structure (S_A, \circ) is a permutation group.

    The group (S_A, \circ) is called the symmetric group on A. Therefore, every symmetric group is a permutation group.

    Question:

    I know a permutation is a group of functions. What makes a group symmetric? Could some one please show me an example?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by novice View Post
    For every nonempty set A, the algebraic structure (S_A, \circ) is a permutation group.

    The group (S_A, \circ) is called the symmetric group on A. Therefore, every symmetric group is a permutation group.

    Question:

    I know a permutation is a group of functions. What makes a group symmetric? Could some one please show me an example?
    The symmetric group S_n is, by definition, the group of all the permutations on n points. These permutations are functions, as you pointed out.

    These groups are called permutation groups because they are the groups which define the symmetry of a regular object with n points. Such an object is best described in (I believe) n-1-dimensions, where it can always be drawn without lines crossing over.

    For examples, S_3 is the group of symmetries of a triangle, while S_4 is the group of symmetries of a tetrahedron.
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