Binary operations and groups

Example 1.3. The examples are almost too numerous to mention. For example, using +, we have (N,+), (Z,+), (Q,+), (R,+), (C,+), as well as vector space and matrix examples such as (Rn,+) or (Mn,m(R),+). Using subtraction, we have (Z,−), (Q,−), (R,−), (C,−), (Rn,−), (Mn,m(R),−), but not (N,−). For multiplication, we have (N, ·), (Z, ·), (Q, ·), (R, ·), (C, ·). If we define Q∗ = {a ∈ Q : a 6= 0}, R∗ = {a ∈ R : a 6= 0}, C∗ = {a ∈ C : a 6= 0}, then (Q∗, ·), (R∗, ·), (C∗, ·) are also binary structures. But, for example, (Q∗,+) is not a binary structure. Likewise, (U(1), ·) and (μn, ·) are binary structures. In addition there are matrix examples: (Mn(R), ·), (GLn(R), ·), (SLn(R), ·), (On, ·), (SOn, ·). Next, there are function composition examples: for a set X, (XX , ◦) and (SX , ◦). We have also seen examples of binary operations on sets of equivalence classes. For example, (Z/nZ,+), (Z/nZ, ·), and (R/2πZ,+) are examples of binary structures. (But there is no natural binary operation of multiplication on R/2πZ.)