Nice intro to connections between algebra and geometry

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Discussion Overview

The discussion revolves around the connections between algebraic structures and geometries, particularly focusing on the need for accessible literature that bridges these concepts for (under)graduate students. Participants explore various questions related to the relationship between geometries and groups, the significance of different types of groups, and the geometric implications of algebraic structures.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant expresses a desire for a book that addresses the connections between algebra and geometry, specifically for students, and lists several questions regarding the nature of these connections.
  • Another participant shares a link to an article that may meet some of the original poster's requirements, though they express uncertainty about its appropriateness for the intended audience.
  • A participant notes that transformations preserving geometric properties form a group and suggests that one can find geometric objects corresponding to abstract groups.
  • A different participant recommends a book titled "Geometries and Groups" by Nikulin and Shafarevich, describing it as elementary yet challenging.
  • Another book, "Glimpses of Algebra and Geometry" by Toth, is mentioned as a recommendation, although the participant has not yet read it.

Areas of Agreement / Disagreement

Participants generally agree on the importance of finding accessible resources that connect algebra and geometry, but there is no consensus on a single recommended book or approach, as multiple suggestions are offered.

Contextual Notes

The discussion highlights the need for literature that addresses specific questions and concepts in a manner suitable for students, indicating a gap in available resources that cater to this audience.

Who May Find This Useful

This discussion may be useful for (under)graduate students seeking to understand the connections between algebra and geometry, as well as educators looking for appropriate teaching materials.

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

Toth's Glimpses of Algebra and Geometry
 
I'm grateful to everyone for the suggestions. I'll take a look at all of them.
 

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