Groups

A group is a pair $(G,\circ )$ where $G$ is a set and $\circ$ is a binary operation on $G$ such that the following properties hold:

• Closure: $a\circ b\in G$ for all $a,b\in G$
• Associativity: $a\circ (b\circ c)=(a\circ b)\circ c$ for all $a,b,c\in G$
• Identity Element: there is an element $e\in G$ such that $e\circ a=a\circ e=a$ for all $a\in G$
• Inverses: for all $a\in G$ there is some $b\in G$ such that $a\circ b=b\circ a=e$

A group is called Abelian if $gh=hg$ for all $g,h\in G$