A group G is a collection of objects with an operation satisfying well-established rules (axioms). Also see Wikipedia’s definition.
Today I realized that the operation is actually what defines a group, NOT the collection of objects. A collection of object is not a group (of objects) as I previously thought. My misunderstanding of the topic was probably because I was making a connection between the English meaning of the word group and the mathematical term group. Also, a group can lose its status if the operation changes but the collection of objects remains the same (if some axioms stop being true for the new operation).