People who know something about math are going to spot several errors in the presentation. For example, the reals with multiplication don't form a group, since 0 has no multiplicative inverse.
Not an error, but a subtlety: the collection of axioms here says that the group has a right identity and right inverses. It's later assumed that the identity is also a left identity (in the proof of uniqueness). This is true, as it happens, but it's not entirely obvious. The usual presentation of groups has two-sided identities and inverses up-front.
Actually, the alternative axiom set of a right identity, but left inverses, has models that are not groups. An example is a left zero semigroup, where ab = a for all a and b. We have right identity (ae = a; no problem) and left inverses (for all a, there exists b such that ba = e; yes, just take b = e). This isn't a group! (OK, unless it has only one element.)
Maybe this illustrates an interesting point for developers: the axiomatic approach is like having a common interface for some class of objects, and it's helpful to be able to reason about them without knowing the details of how they're implemented. But reasoning about the expressive power of different interfaces or specifications can be horribly difficult. The Robbins conjecture (about the equivalence of two axiom sets for Boolean algebra) was open from 1933 to 1996 - and proved in an automated theorem prover, no less.