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
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
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.
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
. Also, even though the order of group
composition usually matters , in the case of inverses, it does not:
x ◦ x
◦ 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
The letter e is traditionally used for the neutral element, probably because it begins the
German word einheit, which means “identity.”