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I understand that this is a bit of offtopic but mathoverflow is my last resort, as google did not help.

I am about to publish an English translation of my Russian book for high school students. The problem is that many of our references are in Russian and not translated. So, I would be very grateful if people can recommend:

1) An introduction to group theory suitable for high-school students. Ideally it would emphasize the idea of the symmetry, discuss permutations in details and maybe prove something like Cayley's theorem in the end.

2) A book on the Fundamental Theorem of arithmetics.

3) An elementary book on Galois theory

4) Something very elementary about topology, like Mobius bundles, classification of surfaces, knots, maybe, a little bit of general topology, like Cantor set.

5) A few books for younger students, that is, books on mathematical puzzles or simpler olympiad problems.

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    $\begingroup$ What do you mean by a book on the fundamental theorem of arithmetic? I can't imagine there is a book entirely about that theorem. I can see from your CV that the book you are translating is about math olympiad problems, so presumably the students who would read your book are not the typical high school students. What kind of background do you imagine your readers would have? $\endgroup$
    – KConrad
    Commented Feb 24, 2011 at 5:00

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Visual Group Theory by Nathan Carter could be used by high school students. It makes great use of Cayley diagrams to show the structure of groups and gently introduces the axiomatic definition of a group in chapter 4 (out of 10).

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Take a look at "Groups and Their Graphs" by Israel Grossman and Wilhelm Magnus. It's part of the Mathematical Association of America's "New Mathematical Library" series of books aimed at high school students. It's the book that first introduced me to the subject. It's accessible to bright high school students and pretty widely available in school libraries.

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The Knot Book by Colin Adams could be useful. It introduces knots and their applications without requiring knowledge of group theory. There's also some introductory material on braids, surfaces and topology.

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  • $\begingroup$ I posted this separately to my Visual Group Theory answer instead of editing that one, as VGT had already received a couple of upvotes and The Knot Book is a very different book. $\endgroup$
    – J W
    Commented Feb 23, 2011 at 12:31
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Regarding 1) I don't know of any good book, but this essay "Group Theory in the Bedroom" by Brian Hayes

http://www.americanscientist.org/issues/pub/group-theory-in-the-bedroom

is elementary and entertaining. A collection of his essays is published in book form under the same title.

Regarding 2), I like 'The Higher Arithmetic' by Davenport.

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1) Hermann Weyl's Symmetry is a classic. ISBN-13: 978-0691023748

4) MA Armstrong's Basic Topology deserves attention. ISBN-13: 978-0387908397

Edit: number (5) has been bugging me so I had to hunt down what was trying to surface in my mind. It is the Berkeley Math Circle, who have books on Olympiad Contest Problems for younger students.

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1,3 and partly 4: Alekseev's "Abel's Theorem" is apparently translated into English.

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Fearless Symmetry by Ash and Gross

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  • $\begingroup$ +1! Wonderful book aggregating various topics in physical systems underlining the theme of symmetry mathematically and visually. $\endgroup$
    – jojo
    Commented Sep 4, 2017 at 4:57
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Unfortunately for Galois theory there isn't anything suitable for high school students, but the nice introduction is here Galois theory for beginners, John Stillwell, in addition to historic essay in the introduction to the book by Edwards.

for arithmetics, I think the books by Alan Baker (Theory of numbers) and by G.H. Hardy might be helpful.

for geometry/topology this essay by S.S. Chern can provide some motivation for the subject.

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  • $\begingroup$ After clicking the link "Galois theory for beginners, John Stillwell", I got the message "Oops! This page appears broken. DNS Error - Server cannot be found". $\endgroup$
    – Victor Makarov
    Commented Feb 23, 2011 at 12:55
  • $\begingroup$ ...and when I clicked it, I ended up at the Chern essay. (Perhaps not so surprising in hindsight, since both links point to the same URI.) $\endgroup$
    – Hans Lundmark
    Commented Feb 23, 2011 at 17:50
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    $\begingroup$ As far as I know, "Galois theory for beginners" is only available if you have access to JSTOR. Also, I have to admit that there are a few small errors in that paper. $\endgroup$
    – John Stillwell
    Commented Feb 23, 2011 at 20:43
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    $\begingroup$ After googling Galois theory for beginners, John Stillwell I downloaded it from cse.unsw.edu.au/~dimitris/galois_theory.pdf $\endgroup$
    – Victor Makarov
    Commented Feb 24, 2011 at 2:12
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4) Victor Prasolov's Intuitive Topology is a translation of an eponymous Russian book (freely avaliable at http://www.mccme.ru/prasolov/ , but only in Russian). It is mostly about knots and homology, although at an elementary enough level not to presume knowledge of groups. I don't know of a good introduction to homotopical topology. Please don't pester western students with general topology; it is already way too popular over here.

1) I can't name a book right out of my head, but there should be some. At the moment I remember Etingof's introduction, but it is probably too compressed to be read by a highschooler without further instruction.

5) Ross Honsberger has many of these.

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  • $\begingroup$ Somehow I did not realize Prasolov has been translated. An interesting point about general topology though. I don't fancy general topology but I believe that at some point any student of math should learn a little bit of it. $\endgroup$
    – Roman Fedorov
    Commented Feb 23, 2011 at 14:25
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    $\begingroup$ My point is that it does not need additional popularity - everybody is going to meet it in the analysis, functional analysis, measure theory and many other courses in university. It is mostly a technical discipline, and learning technique before one knows what this technique is good for is not a good strategy for most people. $\endgroup$
    – darij grinberg
    Commented Feb 23, 2011 at 15:39
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I like Courant and Robbins' book: What is Mathematics? It does not have Galois theory but does describe how to use elementary field theory to prove the impossibility of the three classic Greek construction problems. It also has a section on topology and one on the fundamental theorem of arithmetic. There is a clever proof of that theorem without proving first the usual prime divisibility property.

The book Geometry and the Imagination by Hilbert and Cohn - Vossen, also has some nice elementary topology as I recall.

These books were meant to be accessible to the intelligent lay person.

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I read a book called abstract algebra by dummit and foote second edition. I read it when i was 16 and i found it excellent because it explained everything very well.

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