GROUPS 17

pulled it out, and then rotate it another 45

◦

about the same axis.

Although this exercise will tire you out, the net result is simply

the same element of SO(3) that is described more simply by saying:

rotation by 90

◦

about the point 30

◦

N, 50

◦

W.

Notice how in mathematics we can choose the deﬁnitions to ﬁt

our purpose, in this case to make SO(3) into a group. But after

we announce and agree on a deﬁnition, we have to stick to it until

further notice.

This is all we need to say in order to deﬁne the group SO(3). As

a set, it is the set of all rotations of the sphere, as deﬁned above.

As a group, it has the extra feature of composition: Given any

two rotations x, y we compose them by deﬁning x ◦ y as that single

rotation that has the same effect as doing ﬁrst y and then x.

The General Concept of “Group”

DEFINITION: A group G is a set with a composition deﬁned

on pairs of elements, as long as three axioms hold true:

1. There is a neutral element e in G,sothatx ◦ e = e ◦ x = x

no matter what element of the group is substituted for x.

5

2. For any element x of G, there is some element y in G so

that x ◦ y = y ◦ x = e.

3. For any three elements x, y,andz in G,wehave

(x ◦ y) ◦ z = x ◦ (y ◦ z).

In the second rule, it should be realized that y may possibly

be x itself (e.g., if x = e), but usually it is a different element. In

all cases, we write y = x

−1

. Also, even though the order of group

composition usually matters , in the case of inverses, it does not:

x ◦ x

−1

= x

−1

◦ x = e.

The third rule is called the associative law. It has to hold in

all groups, as it does in SO(3). Why is it true in SO(3)? Because

whether we write (x ◦ y) ◦ z or x ◦ (y ◦ z), we end up doing ﬁrst the

5

The letter e is traditionally used for the neutral element, probably because it begins the

German word einheit, which means “identity.”

Start Free Trial

No credit card required