Introduction

Group theory studies groups.

A group is a set with a single binary operation ($\circ$) that satisfies the four core axioms:

1. Closure. For any two elements $g_{1}\in G$ and $g_{2}\in G$, $g_{1}\circ g_{2}\in G$.
2. Associativity. Grouping does not matter: $(g_{1}\circ g_{2})\circ g_{3} = g_{1}\circ(g_{2}\circ g_{3})$.
3. Identity element. There's a special element in the set (e) where $g \circ e = e \circ g = g$ for any $g$.
4. Inverse element. For every element $g$, there's an inverse element $g^{-1}$ where $g \circ g^{-1} = g^{-1} \circ g = e$.