Group theory is the branch of mathematics that studies symmetry, found in crystals, art, architecture, music and many other contexts. But its beauty is lost on students when it is taught in a technical style that is difficult to understand. *Visual Group Theory* assumes only a high school mathematics background and covers a typical undergraduate course in group theory from a thoroughly visual perspective.

of the technique lies in Step 2, which gives both freedom and restriction: You should look for all manipulations of the object provided that it takes up the same space. The freedom (in fact, the command) to find all manipulations ensures that we get a complete description of the object's symmetry; to do so may require some exploration, as it did in the case of the rectangle. The restriction to manipulations that take up no new space ensures that our exploration pays attention to the object's

35 3.2. Groups of actions Figure 3.14. The arrangement of two couples in a square, ready to begin a square dance or a contra dance. Dancing a figure rearranges the dancers. If they correctly obey the caller, every dance ends with the dancers back to the locations in which they began the dance. Figure 3.15 shows the effects of six example contra dance figures. The particular steps are not shown, only the effect the figure has on the dancers' arrangements. The collection of all such figures

patterns These exercises preview Section 5.2, about an important family of groups called the abelian groups. Exercise 4.22. We saw earlier in this chapter that in the group V4 , the equation RB D BR is true. In fact, for any two elements a; b 2 V4 , the equation ab D ba is true. That is, the order in which you combine elements does not matter. Consider each group whose multiplication table appears in Figure 4.7 (except A5 , whose details are too small to see). For which of those groups does the

will become ambiguous after collapsing to the appearances of those that will not. Collapsing cosets produces ambiguous arrows when some arrows of a given color go from nodes of one left coset to nodes of two or more different left cosets. The left illustration in Figure 7.27 shows this, and it is what caused problems in Figure 7.26. On the other hand, when all the like-colored arrows from any one coset go unanimously to another coset, no ambiguity arises, as the right illustration in Figure 7.27

Four- and higher-dimensional groups exist but are not shown here. n1 ; n2 ; : : : ; nm , such that A Š Cn1 Cn2 Cnm : Although we learned about abelian groups in Chapter 5 and direct products in Chapter 7, we did not encounter isomorphisms until this chapter, and thus were unable to state this theorem until now. I do not include the proof of Theorem 8.8 here, because it is lengthy. I guide interested readers through constructing a proof in the exercises in Section 8.7.10. The important