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Thread: Symmetric Group

  1. #1
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    Symmetric Group

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

    The group $\displaystyle (S_A, \circ)$ is called the symmetric group on $\displaystyle A$. 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 $\displaystyle A$, the algebraic structure $\displaystyle (S_A, \circ)$ is a permutation group.

    The group $\displaystyle (S_A, \circ)$ is called the symmetric group on $\displaystyle A$. 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 $\displaystyle S_n$ is, by definition, the group of all the permutations on $\displaystyle n$ 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 $\displaystyle n$ points. Such an object is best described in (I believe) $\displaystyle n-1$-dimensions, where it can always be drawn without lines crossing over.

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