# Fundamentals of the Theory of Finite Groups

• Guo Wenbin
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 505)

## 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 : AB. 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 ab, ∀a, bA, then f is said to be an injection from A to B; if for any bB, there exists an element aA 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.

## Keywords

Normal Subgroup Finite Group Maximal Subgroup Nilpotent Group Semidirect Product
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