## Gallery

These images are not provided as an educational resource,
but to give you an idea of the textbook's style.
Therefore they are not self-explanatory.
If you are familiar with group theory, you may be able to understand many of
these images with only the brief descriptions provided. If not, or you are
new to group theory visualization, you can learn more from the documentation
that comes with *Group
Explorer* (available online for free).

### The two-lightswitch group (from Chapter 2)

The two-lightswitch group has two generators, the actions of flipping one switch, or the other. This image shows a Cayley diagram for the two-lightswich group, isomorphic to the Klein 4-group.

(Click the image to see a larger version.)

### A molecule of eclipsed ferrocene (from Chapter 3)

Group theory has many applications, in science, art, and many other areas, including other branches of mathematics. One of its applications is the study of the shapes of moleculs and molecular crystals. Chapter 3 teaches how to compute the symmetry group of a molecule like the one shown here.

(Click the image to see a larger version.)

### Symmetries of a frieze pattern (from Chapter 3)

One application of group theory to art is understanding the symmetry in repeated patterns. Frieze patterns like the one in this figure repeat in one dimension. This diagram shows the structure of the frieze pattern's symmetry.

(Click the image to see a larger version.)

### The alternating group *A*_{5} (from Chapters 5 and 10)

The alternating group *A*_{5}, half of all possible
permutations of five items, has 60 elements. Its structure is
integral to the proof that quintic polynomials have no general
solution by basic arithmetic and radicals. This image is a Cayley
diagram of the group, illustrating its stability, structure, and symmetry.

(Click the image to see a larger version.)

### The quotient process (from Chapter 7)

Some groups contains normal subgroups, which can organize the overall
structure of the larger group, and enable simplification of that
structure by taking a group quotient. This figure shows the quotient
process that divides the quaternion group *Q*_{4} by a
subgroup isomorphic to *C*_{2} and obtains a group
isomorphic to the Klein 4-group.

(Click the image to see a larger version.)

### An infinite quotient (from Chapter 7)

The quotient process can also be performed on infinite groups.
In this diagram, the group of integers * Z* is divided
by the subgroup <12> to create a structure in which arithmetic
takes place mod 12. That structure is then shown to be isomorphic
to the group

*C*

_{12}, using a triangle of homomorphisms as given by the Fundamental Homomorphism Theorem.

(Click the image to see a larger version.)

### A homomorphism between multiplication tables (from Chapter 8)

Homomorphisms can be visualized between Cayley diagrams, multiplication
tables, and cycle graphs. This illustration shows an embedding of the
group *C*_{3} in the group *C*_{6} using
dotted arrows between multipliation tables.

(Click the image to see a larger version.)

### Sylow Theory (from Chapter 9)

Sylow Theory provides information about what structures are guaranteed
to exist within a group, based only on the group's order.
In this figure, we see that the fact that the group *A* has order
200 leads to conclusions about groups it has of orders 2, 4, 8, 5, and 25.

(Click the image to see a larger version.)

### Nonabelian group of order 21 (from Chapter 9)

Sylow Theory also helps classify all groups of a given order.
Using the Sylow Theorems to analyze groups of order 21 produces two results,
one abelian group *C*_{21} (isomorphic to *C*_{3}
x *C*_{7}) and one nonabelian group, whose intricate Cayley
diagram is shown here. Notice how Cayley diagrams
communicate even this intricate structure beautifully.

(Click the image to see a larger version.)