I am a math teacher (teaching lower-level math at a university) who is reviewing abstract algebra after 20 years of not thinking much about it. I have amassed quite a number of abstract algebra books over the past few months and have been jumping around between them as I work through this stuff. I have decided to review several of them. So here's what I think of this one:
Gallian does do an admirable job of explaining concepts. There are decent examples. I think the book is a bit heavy on the applications especially near the beginning. Gallian does tend to be a bit wordy at times, but not bad. As I said, he does explain things pretty well. Based only on the actual text, I'd have given the book 4 stars.
So why only 3 stars? Because I gave the exercises only 2 stars. They very nearly enrage me. Okay, it's not so much that the exercises themselves are horrible. My gripe is that they tend to be either routine problems (that's not my gripe) or overly difficult proofs/sophisticated conceptual problems. There's not a lot in between. That's my first criticism. Next, Gallian seems to have made very little attempt to grade the problems in increasing difficulty. You look at number 3 and you're thinking about it for a week, until finally you are tired and just move on to the next section... when all along, the "meat and potatoes" problems are buried in the sorts of problems that tend to take many hours of pondering. In my opinion, that's not a good way to present the problems. Next, there needs to be MANY, many more routine, concept-building problems. The problems that are there are okay, they just need to be tripled in number with the sorts of routine concept-builders that aid in understanding and cementing definitions and methodology. Leave the abstract proofs for the end of the problem sets.
Also, and this is more of a philosophy issue, just put the damn solutions in the back of the book. It's not a secret (or at least it shouldn't be). As an instructor, I have no problem coming up with testing material on my own and I've never picked even problems from the books. In fact, I've never assigned a problem that my students haven't had the answer waiting when they need to confirm their performance. A student not knowing if he/she is doing things correctly is a sure recipe for reinforcement of bad mathematical habits when they are not doing things correctly. They need to see their faults RIGHT AWAY and clear them up. It's stupid to have a student spending weeks not knowing if he/she has the material down and going into a test that way. Immediate feedback is critical, in my opinion. And it's my opinion that all textbooks should have the solutions available. Solve and confirm. Solve and confirm. Solve and confirm. That's what builds understanding, not a mysterious cloud that a student gets lost within. Just because a math genius can rip through a problem set with no feedback, confirmation, or hint, it's no sign everyone should have to do that. Not everyone can paint a Mona Lisa and not everyone can internalize this stuff on first glance. This "mathematical maturity" argument is silliness. One does not attain "mathematical maturity" via secrecy, reinventing the wheel, or discovering all of mathematics on his/her own. And all the blood, sweat and tears spent working a problem or doing a proof is 100% in vain when the student does not know if he/she has done the thing correctly.
Anyway, getting off my soapbox, this book is decent and you could do much worse. But, at the same time, the book could be so much better with just a bit of refinement. So... not bad, but there are others I'm more fond of (which I will review).