 A symmetry is a transformation that maps a figure onto a copy of itself. For example, if we rotate a square 180 degrees, we get a copy of the square. Or if we reflect a square across a vertical line, we get a copy of the square. So we might try to find the symmetries of a square, and so we see that a square has rotational symmetries of 90, 180, and 270 degrees. It also has a vertical reflection symmetry and a horizontal reflection symmetry. Now if we want to create a group from symmetries, we need an associative binary operation, an identity, and an inverse. And since symmetries are transformations, we can use composition of functions as our associative binary operation. So for example, let v be a reflection across a vertical line and r a counterclockwise rotation of 90 degrees. And let's see which of v, r, vr, or rv are symmetries of the isosceles right triangle shown. So if we reflect across a vertical line, we don't get a copy of the figure. If we rotate by 90 degrees, we don't get a copy of the figure. If we rotate by 90 degrees, and then reflect across a vertical line, we do. So vr is a symmetry. If we reflect across a vertical line, then rotate by 90 degrees, we don't get a copy of the figure. And so only vr is a symmetry. It's worth noticing that neither v nor r is a symmetry, but the product vr is, but the product rv is not. So again, in order for a group to exist, we need an identity. And we need a transformation i, where ai is equal to ia is equal to a. So the thing to keep in mind here is things that do the same thing are the same thing. So for ia to be a, we'd have to perform transformation a, and then something else, to have done the same thing as performing transformation a. So in order to do that, we'd perform transformation a, and then do nothing, and we'll have done a. Meanwhile, for ai to be a, again, we'll have to have performed two transformations to have done a, and so we can do that by doing nothing, and then doing a, and that will have done a. And so what this says is if we let i be the do nothing transformation, then the symmetries of an object form a group under composition. Now, since this is a theorem, we should actually prove it. So let's prove closure. Suppose a and b are symmetries. Then ab, remember, that's we're going to do b first then a. So this does b to produce a copy of the object, and then does a to produce a copy of the object. And so that means that ab produces a copy of the object, and so it's a symmetry. And so our set is closed under our binary operation. As for the rest of the group requirements, well, you should do your own homework and prove them yourself. Now, there's only one problem with these symmetries. It would be nice to write a product as another symmetry. But since symmetries produce copies of the object, then all symmetries are the identity. Or are they? We hope not because a group consisting of just the identity is not very interesting. So to see the differences will label the vertices. Now, the labels are not part of the figure, so if it doesn't matter if they change. But this vertex labeling allows us to do two things. We can define symmetries as functions that map vertices to vertices and to define the symmetry group as a group of permutations on the vertices. For example, let v be a vertical reflection, h a horizontal reflection, and r theta to be a counterclockwise rotation through an angle of theta. And let's express hv as a permutation. And what symmetry is hv equal to? So hv means we're going to reflect across a vertical line, then reflect across a horizontal line. Now, since we've labeled the vertices, we see where each vertex gets mapped to. And so that means that this transformation corresponds to the permutation. One gets sent to three. Two becomes four. Three becomes one. Four becomes two. And if we think about this as a symmetry, this is the same as a rotation by 180 degrees. And so hv is r 180 degrees.