All of the groups that we have looked at so far have an additional property that we did not list on page 8:
Groups whose operations have this additional property are called commutative groups or, more frequently, abelian groups. (Why abelian?) Groups whose operations do not have this property are called non-commutative or non-abelian.
Among the important non-abelian groups are the dihedral groups, D_{n}. To illustrate these, let us begin by looking at D_{3}. The elements of D_{3} are movements of an equilateral triangle that result in the triangle's still looking the same as it did before you moved it. Such a movement is called a symmetry. For example, if you flip an equilateral triangle around any one of its altitudes, the triangle still looks the same -- see the first figure below. The pink lines and the labels on the vertices are to help you to see what is going on but are not part of the triangle itself. Therefore, we do not say that the triangle looks different if the only change in its appearance is in the positioning of the labels or the pink lines. There are six different symmetric movements of the triangle; three of these involve flipping the triangle across an altitude as in the following pictures:
Let's call this movement a. | |
Let's call this movement b. | |
Let's call this movement c. |
The other three symmetric movements are rotations though 0 (or 360) degrees, 120 degrees and 240 degrees as in the following figures:
Let's call this movement e. | |
Let's call this movement p. | |
Let's call this movement q. |
The operation we will associate with this set of movements is just to perform the movements one after the other.
For example if you perform movement a: | |
followed by movement p: | |
you get the same result you would have gotten from performing movement c: |
So a * p = c.
We can try this with all combinations to get the multiplication table for this operation:
Notice that properties 1 though 3 hold for this operation on this set of movements, but property 4 does not hold. For example, a * p = c, but p * a = b. They are not the same. Therefore, D_{3} is a non-abelian group. Take a look at the PascGalois triangles for this group.
Of course, the equilateral triangle is not the only shape with symmetries. Although many shapes have symmetries, the easiest family of shapes to study in this manner are the regular polygons. Remember that a polygon is a many sided figure (where the sides are straight edges), and a regular polygon is one whose sides are all the same length. Thus an equilateral triangle is a regular polygon, a.k.a. a regular 3-gon, a square is a regular 4-gon, and the U.S. Pentagon is built in the shape of a regular 5-gon. A square has eight symmetries. You can flip it about either of its two diagonals or about either of its two altitudes or you can rotate it through 0, 90, 180, or 270 degrees. (Remember that a rotation through 360 degrees is the same as a rotation through 0 degrees.) So there are 4 flips and 4 rotations that are symmetries for the regular 4-gon. Thus D_{4} has 8 elements. Similarly, there are n flips and n rotations that are symmetries for a regular n-gon. The rotations are through multiples of 360/n degrees and the flips are across axes formed in the following ways:
If n is even, join the midpoints of sides that are across from each other or join corners that are across from each other. |
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If n is odd, join each corner to the midpoint of the opposite side. |
Thus the symmetries of a regular n-gon will form a group with 2n elements, and we call the group D_{n}. We define the multiplication in the same way we did in the D_{3} case. Again, performing a flip followed by another flip results in a rotation, performing a rotation followed by another rotation results in a rotation, and performing either a flip then a rotation or a rotation then a flip, results in a flip. (You can verify this by cutting out a regular hexagon and a regular pentagon, labeling the corners, and flipping and rotating.) It is possible to generate multiplication tables and thus PascGalois triangles for these groups. Then it is interesting to see what the triangles look like.