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

204 MATHEMATICAL PHILOSOPHY
no integer n such that zXn =n X2 I; and so S2, though
it has the group property, is not a group.
Let S3 be the system consisting of the class C of all the
positive and negative integers including zero and of
addition as the rule of combination; you readily see that
Sa is a group, zero being the identical element, and each
element having its own negative for reciprocal.
A group is said to be finite or infinite according as its
C is a finite or an infinite class and it is said to be Abelian
or non-Abelian according as its rule of combination is or
is not commutative-according, that is, as we have or do
not have aob =boa, where a and b are arbitrary members
of C. You observe that the group S3 is both infinite and
Abelian.
For an example of a group that is finite and Abelian
it is sufficient to take the system S4 whose C is composed
of the four numbers, I, -I, i, -i, where i is v-ri, and
whose rule of combination is multiplication; you notice
that the identical element is I, that I and - are each
its own reciprocal and that i and -i are each the other's
reciprocal.
Let S5 have the same C as S3 and suppose o to be subtraction instead of addition; show that S5 has the group
property but is not a group. Show the like for S6 in
which C is the same as before and o denotes multiplication. Show that S7 where C is the same as before and o
means the rule of division, has not even the group property.
Consider Ss where C is the class of all the rational
numbers (that is, all the integers and all the fractions
whose terms are integers, it being understood that zero
can not be a denominator) and where o denotes +; you
will readily find that Ss is a group, infinite and Abelian.
Examine the systems obtained by keeping the same C