How can rotations and other symmetry operations form a group?

In modern algebra, rotations and other symmetry operations form a group because they satisfy the four fundamental axioms of group theory when acting on a geometric object.12

A set of operations forms a group if it meets these criteria:
* Closure: Combining any two operations (such as two rotations) results in another operation that is also within the set. For example, rotating a plane by angle $\alpha$ followed by angle $\beta$ is equivalent to a single rotation by $\alpha +\beta$.13
* Associativity: The order in which operations are grouped does not change the final result: $\alpha (\beta \gamma )=(\alpha \beta )\gamma$.13
* Identity Element: There exists an operation that leaves the object unchanged, such as a rotation of 0 degrees.13
* Inverse Element: Every operation has an inverse that reverses its effect, returning the object to its original state (e.g., the inverse of a rotation by $\alpha$ is a rotation by $-\alpha$).13

When these symmetry operations preserve the properties of a geometric figure, they are classified as a group of transformations, providing a powerful framework for understanding geometry.13