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

THE GROUP CONCEPT 203
proposed and are sometimes used. The definitions are
not all of them equivalent but they all agree that to be a
group a system must satisfy condition (a).
Systems satisfying condition (a) are many of them on
that account so important that in the older literature of
the subject they are called groups, or closed systems, and
are now said to have " the group property," even if they
do not satisfy conditions (b), (c) and (d). The propriety
of the term " closed system " is evident in the fact that a
system satisfying (a) is such that the result of combining
any two of its members is itself a member-a thing in
the system, not out of it.
Various Simple Examples of Groups and of Systems
that Are Not Groups.-You observe that by the foregoing
definition of group every group is a system; groups, as
we shall see, are infinitely numerous; yet it is true that
relatively few systems are groups or have even the group
property-so few relatively that, if you select a system at
random, it is highly probable you will thus hit upon one
that is neither a group nor has the group property.
Take, for example, the system Si whose class C is the
class of integers from I to 10 inclusive and whose rule of
combination is thatof ordinary multiplication X; 3 X4 = I2;
I2 is not a member of C, and so S1 is not closed-it has
not the group property.
Let S2 have for its C the class of all the ordinary
integers, I, 2, 3,..ad infinitum, and let o be X as
before; as the product of any two integers is an integer,
(a) is satisfied-S2 is closed, has the group property;
(b), too, is evidently satisfied, and so is (c), the identity
element being I for, if n be any integer, n X I = Xn =n;
but (d) is not satisfied-none of the integers (except I)
composing C has a reciprocal in C-there is, for example,