Elementary mathematical books 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.
 A: 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.  
A: 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.
A: 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.
A: 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.
A: 1,3 and partly 4: Alekseev's "Abel's Theorem" is apparently translated into English.
A: Fearless Symmetry by Ash and Gross
A: 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.
A: 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.
A: 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.
A: 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.
A: 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).
