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GROUPS 19
of physics were made through idealized approximations , so that
patterns could be observed, for example, the inclined plane of
Galileo, or the thought experiments of Einstein.
Group theory gives us a mathematical way to deal with sym-
metry. But the actual earth has no exact symmetries: No rotation
(except the neutral, do-nothing rotation) takes it into a position
where it exactly coincides with itself. The idealized sphere, on the
other hand, is very symmetrical. This is reﬂected in the fact that
SO(3) has many different elements—all rotations, which smoothly
turn the sphere within its container.
Digression: Lie Groups
Our i dealized earth turns smoothly all the time at exactly the same
rate. There should be a concept of “inﬁnitesimal rotation”—how
much the earth turns in just an instant of time. This would be an
“inﬁnitesimal” element in SO(3), something like an angular velocity
around a certain axis. Any actual rotation should be buildable by
integrating some inﬁnitesimal rotation. This is the germ of the
theory of Lie groups and Lie algebras, named for the Norwegian
mathematician Sophus Lie (pronounced “lee”). Lie group theory is
thus a kind of marriage between calculus and group theory. It has
a place in number theory too, but that is beyond the scope of our
book.
A group such as SO(3) that has inﬁnitesimal generators is called
a continuous group. The other main type of group is a called a
discrete group.Inthelatter,thereisnosmoothpathfromone
element of the group to another. An example of a discrete group
is the set of integers, where we “compose” two integers by adding
them together. The neutral element is 0 and the inverse of x is x.
The next type of group we will study, a permutation group, is also a
discrete group.
Other examples of Lie groups include the set of rigid motions of
space to itself, used in crystallography and Newtonian physics; the
set of Lorentz transformations of ﬂat spacetime, used in relativistic
physics; the set of rotations of a circle, also called SO(2); and the
20 CHAPTER 2
set of real numbers with composition given by ordinary addition.
(Composition in the ﬁrst three of these examples is given by doing
ﬁrst one motion and then the second, similar to composition in
SO(3).)
Lie groups turn up when we study a geometric object with a lot
of symmetry, such as a sphere, a circle, or ﬂat spacetime. Because
there is so much symmetry, there are many functions from the
object to itself that preserve the geometry, and these functions
become the elements of the group. As we will see, discrete groups
can be also used to keep track of symmetries.

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