We'll throw some light on the title question of this page by asking another question. By taking this abstract approach group theory deals with many mathematical systems at once.
Group theory requires only that a mathematical system obey a few simple rules.
The theory then seeks to find out properties common to all systems that obey these few rules. Any mathematical system that obeys those four rules is a group. The set of negative integers, for example, is not closed under multiplication because the product of two negative integers is not a negative integer. The set of vectors of unit magnitude is not closed under vector addition because the vector sum of two unit vectors is not necessarily a unit vector. On the other hand the counting numbers are closed under addition and multiplication.