> The thing is group theory is taught in an incredibly abstract manner, its hard to find any motivating application for it, or any problems it helps us solve.
Mathematics is abstractly defined. But for basic group theory there's a plentitude of very concrete examples to rely on.
> Also terminology/definitions are vague too,
Absolutely not. There is no vagueness at all! Everything is completely well-defined in most introductory textbooks/courses (or you can even read the precise definitions on Wikipedia, which is often not the case).
> whether a vector has an endpoint or it something unachored in space is itself not clear from many treatments.
Vectors do not have endpoints. Vectors are not anchored. Vectors are elements of vector spaces. Vector spaces are completely clearly defined.
Mathematics is abstractly defined. But for basic group theory there's a plentitude of very concrete examples to rely on.
> Also terminology/definitions are vague too,
Absolutely not. There is no vagueness at all! Everything is completely well-defined in most introductory textbooks/courses (or you can even read the precise definitions on Wikipedia, which is often not the case).
> whether a vector has an endpoint or it something unachored in space is itself not clear from many treatments.
Vectors do not have endpoints. Vectors are not anchored. Vectors are elements of vector spaces. Vector spaces are completely clearly defined.