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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