Groups

Properties

• Closure: For any two elements $a$ and $b$ in $S$, the result of $a * b$ is also in $S$.
• Associativity: For any elements $a, b,$ and $c$ in $S$, the equation $(a * b) * c = a * (b * c)$ holds true. The order of operations doesn't matter.
• Identity: There exists a special element $e$ in $S$ such that for any element $a$ in $S$, $a * e = e * a = a$.
• Inverse: For every element $a$ in $S$, there is a corresponding element $a⁻¹$ in $S$ such that $a * a⁻¹ = a⁻¹ * a = e$, where $e$ is the identity element.
• Commutativity: For any two elements $a$ and $b$ in $S$, $a * b = b * a$. The order of the elements doesn't matter.