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A group is a set $G$ together with a multiplication on $G$ which satisfies three axioms:

  • Associativity: The multiplication is associative, that is to say $(xy)z=x(yz)$ for any three (not necessarily distinct) elements from $G$
  • Identity Element: There is an element $e$ in $G$, called an identity element, such that $xe=x=ex$ for every $x$ in $G$
  • Inverse Element: Each element $x$ of $G$ has an inverse $x^{-1}$ which belongs to the set $G$ and satisfies $x^{-1}x=e=xx^{-1}$