If we want to decide whether a given set of elements with a specific binary operation constitutes a group, we must test to see whether the axioms are satisfied. Let us examine the following sets for eligibility as groups. We begin with Group A (p. 4).

Example 1

Set of elements: All integers (positive, negative, and zero).

Binary operation: Addition.

Associativity: Addition of numbers is associative.

Identity: The set contains zero as an element and, for every integer u, u + 0 = 0 + u = u. Zero is the identity element.

Inverses: If u is an integer, its negative –u is an integer and u + (–u) = (–u) + u = 0; –u is the inverse of u, or, in group notation, u-1 = –u.

Thus, the set under test is a group. Since this group has infinitely many elements, we say the group is infinite. This group will sometimes be referred to as an infinite additive group or the additive group of integers.

Example 2

Let the set be the same as in Example 1, but now consider multiplication. The reader can check for himself that multiplication is a binary operation on the set of all integers and that the axioms on associativity and the existence of an identity element are satisfied. To see if the set satisfies Axiom 3, we try to determine the inverse of the element 2.