18 CHAPTER 2
rotation z, then the rotation y, and ﬁnally the rotation x.Inmany
groups, the associative law is much tougher to check (such as the
matrix groups discussed in chapter 11).
In our deﬁnition, we used the symbol “◦” to stand for the
composition law of elements in our group. We could have used any
notation, such as x + y, x · y, x ∗ y, or even no symbol, as in xy.Each
group has its own law of composition. It can be whatever we deﬁne
it to be—addition, multiplication, whatever—as long as the three
axioms in the deﬁnition hold true.
DEFINITION: If G is a group, the group law istherulethat
tells how to combine two elements in the group to get the
third. We will usually write this combination as x ◦ y,but
occasionally as x + y or even xy.
We will encounter many more examples of groups and group laws
as our journey continues.
In Praise of Mathematical Idealization
The group SO(3) is dear to our hearts. It is around us all the time.
Every second of the day and night, the earth performs the same
rotation in SO(3), if we ignore slight discrepancies in the speed
of the earth’s rotation, and if we ignore the fact that the earth
is not perfectly spherical. It pays to ignore these things for our
purposes, because if we included every bump on every log we would
never see the forest for the trees. That is, we would not be able
to see any patterns that repeat. Every pattern would be ever so
slightly (or more!) different from every other. We could never see
the regularities to abstract from the welter of reality. Mathematics
would never be born.
Even if your interest lies in the realities—whether you are a
physicist, or just an ordinary person going about your daily round of
activities—you have to abstract mathematically. For example, you
use a clock that goes around twice in 24 hours, even though an
astronomical day is not exactly 24 hours long. The great advances