Undergrad Nice intro to connections between algebra and geometry

[SVN]

Connections between algebraic structures and geometries are mentioned in almost any course of modern geometry or algebra. There are monographs dedicated to the subject. Unfortunately, the books, I managed to find, are written for professional mathematicians. I am looking for a book that focuses on those connections written with (under)graduate in mind, with thorough explanations and examples (maybe, not just purely mathematical examples, but ones borrowed from physics), that could answer my questions like the following (I know they are incorrectly posed, too vague, even naïve, but this is exactly why I am in need for such a book, to be able to ask right questions):

- why geometries correspond to groups, but not to fields or to rings (those possesses group structure anyway)?
- properties of groups to be abelian or Lie groups; what does it mean for the geometric properties?
- geometric meaning of existence of normal subgroups and quotients; zero dividers?
- complex numbers are intimately related with geometry; what about quaternions, I know they are used to describe rotations, but this seems to be scanty compared to rich geometric applications of complex numbers; why is that so?

Again, I am not looking for answers for the questions above, but for a book that would help me to dissipate darkness surrounding them.

 

[fresh_42]

Here's a good article which covers a bit of your requirements. Not sure, whether it is at the level you're looking for
https://arxiv.org/pdf/1205.5935.pdf

 

[mathwonk]

basic fact: given a geometric space, such as a euclidean plane, the set of one to one transformations that preserve the geometric properties like distance, form a group. more interesting, given an abstract group, one can search for a geometric object whose set of transformations are identical with this group.

here is a very well written book, elementary in terms of prerequisites, but still fairly challenging:

Geometries and Groups, by Nikulin and Shafarevich.

 

[StoneTemplePython]

A recommended book sitting on my desk, that I have not gotten to yet:

Toth's Glimpses of Algebra and Geometry

 

[SVN]

I'm grateful to everyone for the suggestions. I'll take a look at all of them.