The Theory of Classes of Groups pp 1-57 | Cite as

# Fundamentals of the Theory of Finite Groups

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## 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.

## Keywords

Normal Subgroup Finite Group Maximal Subgroup Nilpotent Group Semidirect Product## Preview

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