Elementary Group Theory

Elementary Group Theory

In mathematics and abstract algebra, a group is the algebraic structure, where is a non-empty set and denotes a binary operation called the group operation. The notation is normally shortened to the infix notation, or even to .

A group must obey the following rules (or axioms). Let be arbitrary elements of . Then:

  • A1, Closure. . This axiom is often omitted because a binary operation is closed by definition.
  • A2, Associativity. .
  • A3, Identity. There exists an identity (or neutral) element such that . The identity of is unique by Theorem 1.4 below.
  • A4, Inverse. For each, there exists an inverse element such that . The inverse of is unique by Theorem 1.5 below.

An abelian group also obeys the additional rule:

  • A5, Commutativity. .

Read more about Elementary Group Theory:  Notation, Alternative Axioms, Subgroups, Cosets

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