# Symmetric Group

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

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.