Examples of group action ↩

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created: 2021-12-21 19:31:42
modified: 2022-01-10 04:13:04

Example 1

Let's consider the point group. It can be represented with the symmetry transformations of a triangle:

Now look at the point on one of its corners:

The orbit of is then:

And the stabilizer of is just , the identity element of the group.

Example 2

Again consider the point group, but now we transform a set of points (orange) instead. and its orbit (blue + orange):

The stabilizer of is again just the neutral element.

Example 3

Again consider the point group, with the set of points .

Here we can see that leaves unchanged, so the stabilizer of is:

Example 4

Now look at the point group, where its elements are the symmetry transformations of a square:

Then we act on the point set and the transformed sets are the following:

From this we can see that and of course leaves invariant, so the stabilizer of is the following:

And the orbit is:

Example 5

Again we use , but the point set is different:

The transformed versions of are:

From this we can see that the stabilizer is:

And the orbit is the union of all of them: