Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

THE GROUP CONCEPT 211
for, if a be any flight, it is plain that aoO =Ooa =a. And
condition (d) is satisfied for it is evident that aoa'=
a'oa =O where a' and a agree in length and direction but
are opposite in sense. Hence the system of angel flights
is a group. And it is easy to see that it is both infinite
and Abelian.
What I have here called an angel flight is known in
mathematics and in physics as a vector; a vector has no
position-it has its essential and complete being in having
a length, a direction and a sense. And so, you see, the
system composed of the vectors of space and of vector
addition as a rule of combination is an infinite Abelian
group.
Connection of Groups with Transformations and Invariants.-Let us have another look at our angel flights, or
vectors. I am going to ask you to view them in another
light. Let V be any given vector-that is, a vector of
given length, sense and direction; where does it begin
and where does it end? A moment's reflection will show
you that every point in the universe of space is the beginning of a vector identical with V and the end of a vector
identical with V. Though these vectors are but one, it is
convenient to speak of them as many equal vectorshaving the same length, direction and sense. Let the
point P be the beginning of a V and let the point Q be
its end. Let us now associate every such P with its
Q(P -> Q); we have thus transformed our space of points
into itself in such wise that the end of each F is the
transform of its beginning; call the transformation T;
let us follow it with a transformation T' converting the
beginnings of all vectors equal to a given vector V' into
their corresponding ends. What is the result? Notice
that T converted P into Q and that T' then converted Q