Abstract Algebra - 1.3 The Dihedral Groups

[@yalirothenberg9194]

Just want to say thanks for these videos. One of my fave go-to math youtubers

[@-Mohamed_bayan]

You have a special way of teaching, Thanks Dr brehm.

[@jakehobrath7721]

Thank you! Dihedral groups are very intuitive, but something was DRIVING ME NUTS when I watched other videos. The rf element, for example, denotes the symmetry created by rotation then a flip. However, when group operation is composition, it is written in reverse. So I always wondered why the element is not written fr. From watching your video, I’ve learned that there is a difference between naming symmetries and the action of compositing them. The associative property makes way more sense now that I understand this.

[@xoppa09]

You solved a puzzle in notation. I also like how the teacher inserts parentheses when she does right to left composition, versus leaving it out for the name of the completed symmetry. e.g. "rf" = (f)(r) , or perhaps "rf " = f ∘ r

[@xoppa09]

I’m writing this with the hope of providing some clarity-so that everything clicks into place. Let’s start by labeling the corners of a square as A, B, C, and D in counterclockwise order, starting from an arbitrary corner (labeling them clockwise is also fine, as long as we remain consistent). We observe that after performing several 'rigid motions,' the relative adjacency of the corners A, B, C, and D is preserved. This means that B is always adjacent to A, C is adjacent to D, and D is adjacent to A, regardless of the specific motion applied. In other words, the symmetries preserve A and C, as well as B and D, at opposite corners. Now, how many ways can we rigidly move a transparent square in three dimensions, so that it fits within its footprint on the ground, and the labels on its corners (A through D) remain the same when viewed from both sides? To answer this, we can use a combinatorial argument to count the possible labelings of the corners. Since we’re setting the square into its own footprint, we begin by considering the four possible corners to which we can move A. When A moves, C must remain fixed, as it is always opposite to A. Once A’s position is chosen, which automatically determines C’s position, we can then place B into one of the remaining two corners. This automatically fixes D, as it must be placed in the remaining corner, which is opposite B. Thus, for each of the 4 possible positions for A, there are 2 possible positions for B, leading to a total of 4 × 2 = 8 symmetries. As mentioned in the video, these symmetries correspond to 4 counterclockwise rotations and 4 reflections. If you follow the sequence of the labels A, B, C, D in any of the 4 rotations, you will see that the letters follow a counterclockwise direction. This happens because we labeled the corners in that order, and rotations preserve orientation. While rotations preserve the counterclockwise orientation, a reflection reverses it, causing the letters to appear in clockwise order (try it yourself!). In fact, we can think of a reflection as a "mirror-image rotation"-a concept explored in literature, such as in Alice’s Adventures Through the Looking-Glass. As I mentioned, for each position that A moves to in the original footprint, if we reflect the square in three-dimensional space along the diagonal A-C, we get the corners in reverse order-clockwise. This gives us the 8 symmetries. If you follow a reflection with any number of rotations, the labels will stay in a clockwise orientation. To return to a counterclockwise orientation, you must reflect again. This explains why the product of two reflections is a rotation, and why the product of a reflection and a rotation results in another reflection. In other words, when you reflect, you enter a 'mirror world,' and to escape, you must reflect again. Drawing further from that example in literature (Alice in Wonderland and Through the Looking Glass), we can imagine there are two worlds: the regular world and the mirror world. Are you in the mirror world? Yes, if you've reflected an odd number of times. Are you in the regular world? Yes, if you’ve reflected an even number of times, or if you haven't reflected at all-rotations don’t affect this distinction. Rotations can’t escape the world; only reflections can. While a reflection of a square is technically a three-dimensional motion (or can be treated as such), it's not necessary to consider 3 dimensions to perform a reflection. We can describe reflections simply as special rearrangements of the labels of the corners, which allows us to stay within two dimensions. Of course, it’s useful to think of reflections in three dimensions for visualization, but it's not essential for understanding. Reflections, in this sense, are 'mirror-world' rotations, and this distinction is crucial to understanding the symmetries of the square.

[@AnubhabGoswami-rx9sx]

Why are the arrows in the outer circle clockwise but the ones in the inner circle are anticlockwise. I understand why it's anticlockwise in the inner circle but couldn't get the reason for the outer one.

[@xoppa09]

To answer this briefly, because when you reflect you enter 'mirror world'. See my comment above for more details.