A Guide to Groups, Rings, and Fields

Fernando Q. Gouvêa

Just published by the MAA in its "Dolciani Guides" series. It exists both as a hardcover book and as an e-book. The second link will allow you to see a "Google preview" of the book.

See below for errata.

If you find more mistakes, please let me know!

Errata

  1. Page 11, line –11: there should be an “if” after “isomorphism”.
  2. Pages 11–12, just at the page break: the empty set is indeed an initial object in the category of sets, but it is not a final object (if X has elements, there can’t be a map from X to the empty set). The final object is the set with one element.
  3. Page 41, right in the middle of the page: “as many of one kind as of the over” should read “as many of one kind as of the other”.
  4. Page 43, second paragraph, line 2: “take at G” should just be “take G”. And I don’t actually want GL(2,R), but rather GL+(2,R), the group of invertible real matrices with positive determinant. Real matrices with negative determinant swap the upper and lower half-planes.
  5. Pages 52 and 53. I tried to be careful about the conjugation action, in which the acting group G is the same as the set X on which it acts. But I wonder whether some changes might be needed. For example, on page 52, line –5, maybe I should have written “for any x ∈ X = G”. Something similar is definitely need on page 53; I think I would change line 1 to include “and let x be an element of X = G.”
  6. Page 53, line –6: I should have said “let X be a finite set on which G acts”; the case of X empty is actually not a problem, though it's also not very interesting.
  7. Page 54, theorem 4.6.11, item 3: this is wrong as stated; for example, consider what happens if X is empty! One correct way to say what I mean is this: if g is in Φ(G) and X generates G, then X-{g} also generates G.
  8. Page 54, theorem 4.6.11, item 4: there's probably a space missing between “[Burnside Basis Theorem]” and the statement. Or maybe I just need an italic correction?
  9. Page 59, just below the exact sequence into which the product of two groups fits: The second dotted arrow is certainly a section, but the first is a retraction.
  10. Page 59, line –2: "This one of…" should be "This is one of…"
  11. Page 60, just below the first displayed equation: it is ps, of course, that is the identity; one starts at G2, follows s to G, then returns via p. That's my punishment for writing functions on the left.
  12. Page 61, last lines of the paragraph after Theorem 4.8.5, I say something is either a semidirect product or a direct product of two cyclic groups. It might be objected that it is always a semidirect product, since a direct product is a special case of semidirect product. I'm of two minds about this, but it has confused some readers. Would it be better to rephrase it as “it either is a direct product or is a semidirect product”?
  13. Page 87, Theorem 4.13.10, item 2: I've switched 2 and 3! It is PSL(2,F3) that is isomorphic to A4. It might have also been nice to mention the famous isomorphism between PSL(3,F2) and PSL(2,F7).
  14. In section 4.14.7, it would have been useful to give the standard example of two groups with the same character table, namely D4 and the quaternion group Q.
  15. Page 102, Theorem 4.14.29: the definition of H makes no sense as it stands: what is h? It should be g, of course.
  16. Page 126: in the diagram occurring in item 4, I think, instead of M1/N it should be M1/Ker(f). Either than or say N-Ker(f).
  17. Page 133, beginning of 5.4.3: I say: “Since we can make arbitrary sums and products of modules, we can also contruct direct and inverse limits.” This won’t satisfy people who know category theory. They would prefer something like “Since we can make arbitrary sums and products of modules as well as coequalizers and equalizers of pairs of coterminal homomorphisms of modules, we can also contruct direct and inverse limits.”
  18. Page 150, definition 5.7.21: KX should be K[X].
  19. Page 160, first sentence in section 5.9: There is an extra “a”; the correct sentence is “localization is both…”
  20. I love this one. On page 214, following definition 5.6.13, I make a little speech about how one should always say “a gcd” and not “the gcd”. Then read my next sentence!
  21. Page 230, remark between 6.1.23 and 6.1.24, last sentence: As stated this is clearly nonsense. Isomorphism with what? What I think I meant was that if E and F are linearly disjoint and finite dimensional over K, then their tensor product (over K) is isomorphic to their compositum in L, which really boils down to saying that E tensor F is already a field.
  22. Page 240, line –5: The line should start "Because the elements of a transcendence basis…"
  23. Page 269: There should be only one period after the last sentence of 6.7.10.
  24. Page 280, Reference 60: "Cateogries" should be "Categories".