# Symmetric Group

• Aug 15th 2010, 03:50 PM
novice
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?
• Aug 16th 2010, 12:09 AM
Swlabr
Quote:

Originally Posted by novice
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.