A group records reversible operations that can be combined consistently. Group theory studies their subgroups, group homomorphisms, group actions, and quotient groups. A symmetry group makes the abstraction visible: transformations may change individual points while preserving a specified object or its structure. An orbit of a group action records which positions can be exchanged, while a stabilizer subgroup records which operations leave a selected feature fixed.
Group theory is a branch of mathematics that studies algebraic structures known as groups. A group is defined as a set equipped with a single binary operation that satisfies four fundamental properties: 1. **Closure**: If \( a \) and \( b \) are elements of the group, then the result of the operation \( a * b \) is also in the group.
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