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.
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 subgroup is also called permutation group.
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?