Group Theory: Important Definitions and Results

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• A non-empty set $G$ with binary operation * is called group if the binary operation * is associative and
  • (1) for all $a\in G$, $\exists$ $e\in G$ s.t $a\text{*} e= e\text{*} a =$
  • (2) for each $a\in G$, $\exists$ $a^{-1}\in G$ s.t $a\text{*} a^{-1}=a^{-1}\text{*} a =e$.
• In a group $G$, there is only one identity element.
• In a group $G$, the inverse of the element is unique.
• Every element of $A_n$ is a product of 3-cycles, $n\geq 3$.

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