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

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