Invariance and symmetric groups

Question

My (naïve) idea of symmetry is that it requires some form of invariance under transformation, but I struggle to see how this survives the idea of the symmetric group $ S_n$.

Beyond the bijective property what, if anything, is invariant in a permutation? And if the answer is nothing, what separates a permutation from, say, the rotation of an equilateral triangle through an arbitrary angle?

Answer 1

For an equilateral triangle you can exchange any two points and it will still look the same. You could do the same thing with a tetrahedron, but you need 4 dimensions.

With a square (4 points, 4 sides) it looks different if you swap two adjacent corners without also swapping the opposite corners. However if you also had the diagonals drawn for the square, then you could, as the points and all the connections would stll look the same.

The groups of symmetries generalise this.

Imagine a number of points arranged equally spaced around a circle. Suppose there are lines joining every pair of points. If you swap any two points (moving the lines appropriately) it will still look the same.

You actually don't need any particular arrangement, but the circle may make it a little easier to visualise.

It is the collection of points that remains invariant. If you rotate a triangle through an arbitrary angle the vertices will probably not occupy the same 3 places.

Answer 2

In the $24$ possible permutations of the letters in "WORD" the letters stay the same.

If you have an equilateral triangle in the plane there are three rotations leading to permutations of the vertices. If you allow reflections then all six possible permutations of the three vertices can be accomplished with a rigid motion. In these example it's the set of vertices that is invariant.

If you have a square in the plane the set of permutations of the vertices that can be realized by a rigid motion has either four or eight elements depending on whether or not reflections are allowed.

In general, you get a symmetry group by starting with a set (finite or not) and considering the subset of the set of permutations of that set that preserve some property you are interested in. So every symmetry group is a subset of a symmetric group.

In fact every group is a subset of the symmetric group on its underlying set. That's Cayley's Theorem.