Elementary group theory

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In mathematics and abstract algebra, a group is the algebraic structure \{G,\perp\}, where G is a non-empty set and \perp denotes a binary operation \perp:G\times{}G\rightarrow{}G, called the group operation. The notation \perp(x,y) is normally shortened to the infix notation x\perp{}y, or even to xy.

A group must obey the following rules (or axioms). Let a,b,c be arbitrary elements of G. Then:

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

An abelian group also obeys the additional rule:

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