Introduction to Abstract Algebra I MATH 321 Fall 2025

Consider the symmetries of a square. By this we mean a transformation that maps the square to itself. We find that there are a total of eight such symmetries: four rotations and four reflections (or mirrors or flips).

Let \(r_0\text{,}\) \(r_1\text{,}\) \(r_2\text{,}\) \(r_3\) be rotations counter-clockwise by 0, 90, 180, and 270 degrees (respectively).

We will use \(f_1\) for a reflection (flip) over the horizontal axis (through opposite sides of the square), \(f_2\) for a vertical reflection (through the line bisecting the top and bottom edges of the square), \(f_3\) for a flip over the line \(y = x\) (from the bottom left to top right of the square) and \(f_4\) for the flip over the other diagonal (bottom right to top left).

We notice that when you compose two symmetries, you still get a symmetry. For example, if you first perform the \(f_1\) symmetry, and then \(r_1\text{,}\) this gives you the same result as performing the single symmetry \(f_3\text{.}\) We will write this as \(r_1f_1 = f_3\text{.}\) Notice that we think of this as \(r_1(f_1(□))\text{;}\) we perform the transformation on the right first, then the one of the left.

Here is the table for the group of eight symmetries of the square. This is called the Dihedral Group on eight elements, which we will write as \(D_4\text{.}\)