1. 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?

2. Originally Posted by novice
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.