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One of my professors this semester very graciously offered to give me some projects to work on over the summer, and I happily accepted his offer. The only (minor) drawback is that my interests lie in topology, while his lie in Sylow subgroups and subgroup lattices (among other group theoretic topics). Algebra is my weakest subject, so it will be great to work more in it. However, I was wondering if anyone knew of any connections between Sylow subgroups and techniques in algebraic topology so that I could have some extra motivation when working through the various problems he gives me this summer!

I truly appreciate any insight you may be able to offer!

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    $\begingroup$ There are many connections between group theory and the fundamental group (what is in a name). It is a fact that any finitely presented group occurs as the fundamental group of a CW complex of a two dimensional space. Maybe you can explore how/when the Sylow subgroups arise as fundamental groups of certain subspaces? $\endgroup$
    – Thomas Rot
    May 15, 2018 at 5:54
  • $\begingroup$ Thank you @ThomasRot! This would be a great way for me to explore the connection between the two areas in a way that I could grasp quickly and play around with! $\endgroup$
    – gez460
    May 15, 2018 at 13:07
  • $\begingroup$ Note that I do not know anything about Sylow subgroups, so I don't know if this really leads to anything very interesting. You can also look at a similar problem in other categories (e.g. any finitely presented group occurs as the fundamental group of a four dimensional manifold and you can try to play the same game.) Another topic could be to look at the the correspondence between covering spaces and fundamental groups and see what this implies for the sylow subgroups $\endgroup$
    – Thomas Rot
    May 15, 2018 at 13:26
  • $\begingroup$ I think at the very least it will help build my intuition with Sylow subgroups (along with fundamental groups and covering spaces), which is always a good thing! It will be a perfect place to start! $\endgroup$
    – gez460
    May 15, 2018 at 13:45

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You could take a look at Homotopy properties of the poset of nontrivial p-subgroups of a group by Daniel Quillen and the many later papers citing this article. Alternatively find a chapter in Stephen D. Smith's book Subgroup Complexes that interests you. Finally you could also look at the work of Dave Benson by either picking an article from his Groups, Representations and Cohomology Preprint Archive or a chapter from his books titled "Representations and Cohomology".

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  • $\begingroup$ Thank you for all of the wonderful resources! I can't wait to delve into them and hopefully tell my professor about them as well (maybe we'll be able to study some of the papers--that would be wonderful)! I really appreciate you taking the time to link to all of the resources as well--thank you so much! $\endgroup$
    – gez460
    May 15, 2018 at 13:10

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