# 1

## Introduction to Groups

*Love and Math*

*Symmetry*

## Symmetries of a Square

^{ā}rotation and a 450

^{ā}rotation as equal, since they have the same net effect on every point. With this simplifying convention, it is an easy matter to achieve our goal.

^{ā}followed by a flip about the horizontal axis of symmetry.

*D*. This observation suggests that we can compose two motions to obtain a single motion. And indeed we can, since the eight motions may be viewed as functions from the square region to itself, and as such we can combine them using function composition.

*H*$$ =

*D*because in lower level math courses function composition $$ means ā

*g*followed by

*f*.ā The eight motions

*R*

_{0},

*R*

_{90},

*R*

_{180},

*R*

_{270},

*H*,

*V*,

*D*, and

*D*

^{ā²}, together with the operation composition, form a mathematical system called the

*dihedral group oforder 8*(the order of a group is the number of elements it contains). It is denoted by

*D*

_{4}. Rather than introduce the formal definition of a group here, letās look at some properties of groups by way of the example

*D*

_{4}.

*operation table, the Cayley table*(so named in honor of the prolific English mathematician Arthur Cayley,bio]Cayley, Arthur who first introduced them in 1854) for

*D*

_{4}below. The circled entry represents the fact that

*D = HR*. (In general,

_{90}*ab*denotes the entry at the intersection of the row with

*a*at the left and the column with

*b*at the top.)

*A*and

*B*are in

*D*

_{4}, then so is

*AB*. This property is called

*closure*, and it is one of the requirements for a mathematical system to be a group. Next, notice that if

*A*is any element of

*D*

_{4}, then $$. Thus, combining any element

*A*on either side with

*R*

_{0}yields

*A*back again. An element

*R*

_{0}with this property is called an

*identity*, and every group must have one. Moreover, we see that for each element

*A*in

*D*

_{4}, there is exactly one element

*B*in

*D*

_{4}such that $$. In this case,

*B*is said to be the

*inverse*of

*A*and vice versa. For example,

*R*

_{90}and

*R*

_{270}are inverses of each other, and

*H*is its own inverse. The term

*inverse*is a descriptive one, for if

*A*and

*B*are inverses of each other, then

*B*āundoesā whatever

*A*ādoes,ā in the sense that

*A*and

*B*taken together in either order produce

*R*

_{0}, representing no change. Another striking feature of the table is that every element of

*D*

_{4}appears exactly once in each row and column. This feature is something that all groups must have, and, indeed, it is quite useful to keep this fact in mind when constructing the table in the first place.

*D*

_{4}deserves special comment. Observe that

*HD $$ DH*but $$. Thus, in a group,

*ab*may or may not be the same as

*ba*. If it happen...