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

202 MATHEMATICAL PHILOSOPHY
member of the class can be combined with any member
of it (either with itself or any other member). For a
simple example of such a system we may take for the class
the class of ordinary whole numbers and for the rule of combination the familiar rule of addition. You should note
that there are three and only three respects in which two
systems can differ: by having different classes, by having
different rules of combination, and by differing in both
of these ways.
The definition of the term " group " is as follows.
Let S denote a system consisting of a class C (whose
members we will denote by a, b, c and so on) and of a rule
of combination (which rule we will denote by the symbol o,
so that by writing, for example, aob, we shall mean the
result of combining b with a). The system S is called a
group if and only if it satisfies the following four conditions:
(a) If a and b are members of C, then aob is a member
of C; that is, aob = c, where c is some member of C.
(b) If a, b, c are members of C, then (aob)oc =ao(boc);
that is, combining c with the result of combining b with a
yields the same as combining with a the result of combining c with b; that is, the rule of combination is associative.
(c) The class C contains a member i (called the
identical member or element) such that if a be a member
of C, then aoi=ioa=a; that is, C has a member such
that, if it be combined with any given member, or that
member with it, the result is the given member.
(d) If a be a member of C, then there is a member a'
(called the reciprocal of a) such that aoa' =a'oa =i; that
is, each member of C is matched by a member such that
combining the two gives the identical member.
Other definitions of the term "group" have been