Part I : Groups and Subgroups

We define isomorphic Binary Structures. Main point is, if two binary structures are isomorphic, then propertes of one translate over to properties of the other, via the isomorphism. So, if we know one we know the other. We do not have to study two of them seperately. Definition 3.1. By a binary structure 〈S, ∗〉 , we mean a set S with a binary operation ∗ on it. Definition 3.2. Let 〈S, ∗〉 and 〈T, ∗′〉 be two binary structures. 1. A map φ : S −→ T is called (a map of) or a homomorphism of binary structures if φ(x ∗ y) = φ(x) ∗ φ(y) ∀ x, y ∈ S. 2. A map φ : S −→ T is called an isomorphism of binary structures if φ(x ∗ y) = φ(x) ∗ φ(y) ∀ x, y ∈ S. and if φ is a bijection. (Emphasis in this section is on isomorphic structures; not on homomorphisms) Example 3.3. Let U = {z ∈ C : |z| = 1} be the unit circle. Then, with usual multiplication, 〈U, ·〉 is a binary structure. On the interval [0, 2π) the addition "modulo 2π provides a binary structure ([0, 2π),+). The map φ : [0, 2π) −→ U defined by φ(t) = e is an isomorphism of binary structures.