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Fearless Symmetry by Robert Gross, Avner Ash

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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.”

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