Algebra problem -symmetry group

[dingo_d]

Homework Statement

I have a group of permutations $$S_2$$, which is represented with operators {I, P} that mirror the plane $$\mathbb{R}^2$$ around x=y line. I have to show that this group is fully reducible by constructing invariant subspaces that span $$\mathbb{R}^2$$ that is:
$$\mathbb{R}^2=V_1\oplus V_2$$.

The Attempt at a Solution

I have no idea how to find these subspaces :\ Since I have $$\mathbb{R}^2$$ I could represent that by (x,y) all points, and then try with that, but I'm completely lost :\ Any recommendation what book to look?

 

[mighty2000]

Hello,

I have experience with group theory and I would be happy to help you with this problem.

First, let's define the group S_2 as the group of reflections in the x=y line in \mathbb{R}^2. This group has two elements: the identity element I, which leaves all points unchanged, and the reflection P, which swaps the x and y coordinates of all points.

To show that this group is fully reducible, we need to find two invariant subspaces V_1 and V_2 that span \mathbb{R}^2 and are also invariant under the action of S_2.

One way to find these subspaces is to consider the eigenvectors of the reflection operator P. Since P swaps the x and y coordinates, its eigenvectors will be vectors with equal x and y coordinates. These vectors will be invariant under the action of P, meaning that when we reflect them, they will remain unchanged.

So, let's take the eigenvectors (1,1) and (1,-1). These vectors span \mathbb{R}^2 and are also invariant under the action of P. Therefore, we can define V_1 as the subspace spanned by (1,1) and V_2 as the subspace spanned by (1,-1).

Now, to show that these subspaces are also invariant under the action of S_2, we need to show that I and P leave these subspaces unchanged. This is easy to see, since I leaves all points unchanged and P simply swaps the coordinates, which does not affect the span of these subspaces.

Therefore, we have shown that \mathbb{R}^2 can be decomposed into two invariant subspaces V_1 and V_2, which span \mathbb{R}^2 and are also invariant under the action of S_2. This proves that the group S_2 is fully reducible.

I hope this helps! If you need more help or clarification, please don't hesitate to ask. I would also recommend looking into books on group theory, such as "Group Theory" by M.A. Armstrong or "Abstract Algebra" by David S. Dummit and Richard M. Foote.