The rational numbers are an extension of the integers in which each non-zero number has an inverse under multiplication. Those matrices which do not have multiplicative inverses are called singular. We are looking for the member of the group that x could be replaced by to satisfy the equation. Using the closure axiom and the axiom for inverses we operate on both sides of the equation by the inverse of a . This is the power of abstraction.
Group theory is a clear example of abstraction in modern mathematics.
The emphasis is not on the particular group we might be interested in for some given application. The emphasis is on the basic quality of groupness that all groups have in common. Once a result is demonstrated to be valid for all groups then it is clearly valid for any specific group we may choose. The Little Axiom that Wasn't There Noticibly missing from our list of four axioms is the commutative property. This is because in group theory commutativity is not assumed.