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I'm taking an honors-level algebra class and we're getting into group actions, Sylow subgroups and semidirect products. These topics are fairly intimidating, but the lengthier discussions in the book (Dummit and Foote) have been very helpful in bringing these topics down to earth - for example, investigating the structure of groups of order $pq$, $p^2q$, classifying all groups of order 30, etc.

Seeing these ideas put to use "in the wild" makes them a lot more tractable. Therefore I'm looking for further reading material in this vein, i.e. substantial discussions that use many of the techniques and ideas that one would find in a solid undergraduate group theory course. Ideas I had: research papers in group theory which would be accessible to someone at my level, materials used in graduate courses, miscellaneous articles and expository writings. D&F has plenty of exercises but even more couldn't hurt. And I'm open to sources that go a little beyond what might be expected of an undergraduate.

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  • $\begingroup$ A specialised group theory textbook (as opposed to a general algebra one) will obviously cover it in more depth. I like the book of Roman, but there are many others. $\endgroup$
    – fkraiem
    Commented Dec 1, 2015 at 8:22
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    $\begingroup$ Isaac's book is one of the most beautiful books on finite groups I have ever read. There you can find a lot of theoretic exercises. $\endgroup$
    – Crostul
    Commented Dec 1, 2015 at 8:24
  • $\begingroup$ One project you could undertake that would put a lot of these ideas into action: prove that if $G$ is a nonabelian simple group with order less than $1000$, then $|G|$ must be $60$, $168$, $360$, $504$, or $660$. In other words, use Sylow theory, group actions, etc. to rule out all other possible orders. To quote from Isaacs' Algebra: "The reader is urged to attempt this project, but be warned that the number $720$ is perhaps an order of magnitude more difficult to eliminate than are any of the others." $\endgroup$
    – user169852
    Commented Dec 1, 2015 at 8:30
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    $\begingroup$ I agree with the recommendation of Isaacs' Finite Group Theory. I am working through it myself at the moment, and it is one of the most beautiful math books (not just group theory books) that I have read. His Algebra is also excellent. Warning: after reading Isaacs' smooth, clean exposition, Dummit and Foote will make you want to tear your eyes out. (But D&F has a lot of great examples and exercises, so it is still well worth reading.) $\endgroup$
    – user169852
    Commented Dec 1, 2015 at 8:40
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    $\begingroup$ @Crostul I agree with your recommendation, and so do apparently at least two others. Perhaps you should turn that comment into an answer? $\endgroup$
    – Marc
    Commented Dec 1, 2015 at 10:43

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I recommend "Topics In Algebra" by I.N.Herstein. The exposition there is very clear, and the reading is quite effortless.

The book also contains plenty of great examples & exercises. You actually develop some of the results in the exercises, which is very nice.

(Note that it covers not only group theory).

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There is a book with that very name: Topics in Group Theory by Smith & Tabachnikova. It starts from the basics, more quickly than in most introductory undergraduate abstract algebra texts, and then moves on to a sampling of topics. (For the preface and table of contents, see https://link.springer.com/content/pdf/bfm%3A978-1-4471-0461-2%2F1.pdf)

As you can read in the preface, the title is actually a homage to Herstein's book. It is also supposed to bridge the gap between a first look at group theory and books like Rotman's graduate text An Introduction to the Theory of Groups.

It also contains both exercises and answers.

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