What kind of “symmetry” is the symmetric group about?

There are two concepts which are very similar literally in abstract algebra: symmetric group and symmetry group. By definition, the symmetric group on a set is the group consisting of all bijections of the set (all one-to-one and onto functions) from the set to itself with function composition as the group operation. When the set is finite, the group is sometimes denoted as $S_n$.

The Dihedral group $D_n$, which is a special case of the symmetry group, has a very strong geometric intuition about symmetry as the picture shows.

enter image description here

I know nothing about the relation between these two concepts but the fact that $D_3$ and $S_3$ are actually the same. For me, symmetric group is more about “permutations“. And actually its subgroups are also called permutation groups.

Here are my questions:

  • What’s the relation between these two concepts: “symmetric group” and
    “symmetry group”?
  • What kind of “symmetry” is the symmetric group about?
  • Where is the name “symmetric group” from?

Answer

Mathematically, a symmetry is a bijection of a set onto itself. In some cases, we restrict to bijections that preserve some structure (e.g. isometries, which preserve distances), but in the case of the symmetric group $S_n$ we just have a set with $n$ elements, and no additional structure to preserve, so all bijections (permutations) are allowed.

Attribution
Source : Link , Question Author : Community , Answer Author : Robert Israel

Leave a Comment