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I would like to know more about group cohomology. I know that there are chapters about group cohomology in some group theory textbooks, for example in Rotman's. However my PhD asvisor told me that he did not really like the way in which it was presented, but could not give my any other reference. I was also wondering if there is some text completely devoted to an introduction to group cohomology...

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    $\begingroup$ I'm pretty fond of this book, but I'm not an expert so I'll just leave this as a comment. $\endgroup$ Mar 18, 2018 at 20:05
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    $\begingroup$ Benson, representations and cohomology vols 1&2 is worth a go. $\endgroup$ Mar 18, 2018 at 21:40
  • $\begingroup$ Serre is always good $\endgroup$
    – D_S
    Mar 18, 2018 at 22:43
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    $\begingroup$ I like Brown's "Cohomology of Groups", particularly for how it explains connections with homology-cohomology of topological spaces. $\endgroup$
    – Lee Mosher
    Mar 18, 2018 at 23:36
  • $\begingroup$ @NoahSchweber the book you link at is Gille-Szamuely Central Simple Algebras and Galois Cohomology (always better when we don't have to click the link to follow the discussion). Also D_S probably refers to Serre's Galois cohomology. Your translator, $\endgroup$
    – YCor
    Mar 20, 2018 at 19:31

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Mentioned early Brown's Cohomology of groups is probably best as introductory reference

There are some books which I like and are not widely known.

  • Adem, Milgram, Cohomology of finite groups — contains rarely mentioned in most basic group theory textbooks Quillen-Venkov and Kan-Thurston theorems, and applications in number theory (Brauer grous etc.)

  • Karpilovsky, Group representations, p. 2 — half of book is devoted to detailed analysis of second cohomology group and its properties

  • Stammbach, Homology in Group Theory — here extensions with abelian kernel within a given group variety are discussed

  • Gruenberg, Cohomological Topics in Group Theory and

  • Gruenberg, Categories of group extensions — in some sense, more accessible (and extended) versions of previous text

  • Ю. Кузьмин, Гомологическая теория групп — unfortunately no English translation exists, but you can look at formulas in ch. 4 §8 and ch. 5 §7 or try it with translator (mathematical texts are not very hard to translate usually)

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London mathematical society lecture note series 252 edited by Kropholler + Niblo + Stöhr

Several good articles with concrete applications. (Well, comparatively concrete.)

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