Examples of groups: We now start to survey a wide range of examples of groups (labelled by (A), (B), (C), . . . ). Most of these come from number theory. In all cases, the group axioms should be checked. This is easy for almost all of the examples, and will be left as an exercise except in the occasional more difficult or subtle case.
(A) Our first examples are groups of numbers under addition. To begin, each of the sets Z (the integers), Q (the rational numbers), R (the real numbers) and C (the complex numbers) forms a group under the binary operation + of addition (exercise). Clearly, the groups are all abelian.
(B) For any fixed n ≡ Z, the set nZ = {na : a ≡ Z} is a subgroup of Z (exercise). A few specific cases are:
0Z = {0};
1Z = ( -1)Z = Z;
2Z = ( -2)Z = {2a : a ≡ Z}
= the set of even integers: