The group D4 acts as a group of permutations of the square...

The group $D_{4}$ acts as a group of permutations of the square regions shown below. (The axes of symmetry are drawn for reference purposes.) For each square region, locate the points in the orbit of the indicated point under $D_{4} .$ In each case, determine the stabilizer of the indicated point.

Key Concepts

Dihedral Group

The dihedral group is the group of symmetries of a regular polygon, including rotations and reflections. It encapsulates the structure of symmetry operations that preserve the shape of the object, and in the context of squares, it typically consists of eight elements that perform rotations by multiples of 90° and reflections across various axes.

Group Action

A group action is a formal way of describing symmetries in terms of a group operating on a set. It assigns to every group element a function that maps elements of the set to other elements in a way that respects the group structure. This concept is key for understanding how the elements of the dihedral group can transform points or regions within a geometric figure.

Orbit

The orbit of an element under a group action is the set of points that can be reached by applying all the elements of the group to that particular element. It encapsulates the idea of symmetry by showing all the positions that a point can move to under the group's actions, highlighting the full symmetry pattern.

Stabilizer

The stabilizer of an element in a group action is the subgroup consisting of all the group elements that leave the element unchanged. It is a fundamental concept that helps analyze symmetry properties, as it shows the symmetries that are preserved for a specific point even though the overall group might act more broadly.

Transcript

00:01 So we'd like to find the symmetry group of a square, which i'm actually going to do in a way different from how the hint recommends.

00:10 I'd like to draw a square like this, label the sides 1, 2, 3, and 4, and consider what symmetry sheets we're going to have.

00:25 So i'm going to claim that, copy this.

00:35 I copied the wrong thing.

00:38 I'm just going to duplicate this a couple of times.

00:41 Oh, wait, that's not going to go really nicely for me, is it? i'm going to make just a couple observations real quick.

00:51 First of all, the identity has to be a symmetry, of course.

00:54 And we can also flip it around the horizontal axis.

00:57 We get 1, 2, 3, 4.

01:02 This is a possible symmetry, as well as our standard 1, 2, 3, 4, the identity.

01:11 And then i'll claim, rather than copying this, in general, 1 has to show up somewhere.

01:24 So if i draw a ton of squares here, 1 has to show up somewhere.

01:32 So it can be on the right here, as i have it.

01:34 Or it can be on top.

01:36 Or it can be on the left.

01:40 Or it could be on the bottom.

01:44 Like that.

01:45 1 has to be in one of those places.

01:47 And 2 has to be adjacent to 1, if it's going to be a group.

01:51 We can't go 1, 2, because 2 is next to 1, which means 2 can either be on this side, and we go 1, 2, 3, 4.

01:58 Or it can be on this side, 1, 2, 3, and 4.

02:03 Now i get 1, 2, 3, 4.

02:07 Or going this way, 1, 2, 3, 4.

02:09 I'm going to draw the sides and these squares in a minute, so that we don't 3, 4, so that we can see what's actually going on.

02:23 So here is our base square.

02:28 It's not going to be actually all that square -shaped, because i eyeballed this.

02:31 But it'll think of it close enough.

02:35 So here we have our identity square.

02:43 And we have our various other squares.

02:52 Again, the idea is that vertex 1 has to end up at one of the four vertices.

02:56 So we have four choices there.

03:00 And vertex 2 has to be adjacent to vertex 1, which leaves us two choices for each of those.

03:05 And we get eight possible symmetries...

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