Fundamentals of the Theory of Finite Groups

Abstract

Definition 1.1.1 Let A and be two sets. If to any element a of A, a unique element b of B is assigned according to a certain rule ϕ, then ϕ is said to be a map from A to and B is written as ϕ : A → B. The element b is called the image of a under ϕ and is denoted by b = ϕ(a). The element a is called an inverse image of b under ϕ. Let f be a map from A to B. If f(a) ≠ f(b) for a ≠ b, ∀a, b ∈ A, then f is said to be an injection from A to B; if for any b ∈ B, there exists an element a ∈ A such that f(a) = b, then f is said to be a surjection from A to B. If a map f is both an injection and a surjection, then f is said to be a bijection.