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Are there classes of infinite groups that admit Sylow subgroups and where the Sylow theorems are valid ?

More precisely, I'm looking for classes of groups $\mathcal{C}$ with the following properties:

  • $\mathcal{C}$ includes the finite groups
  • in $\mathcal{C}$ there is a notion of Sylow subgroups that coincides with the usual one when restricted to finite groups
  • Sylow's theorems (or part of them) are valid in $\mathcal{C}$

An example of such a class $\mathcal{C}$ is given by the class of profinite groups.

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Silly example: $\mathcal C$=profinite groups. Sylow subgroups = maximal closed pro-$p$ subgroups. One reason this is silly is that topological groups are not really a type of group. –  Will Sawin Oct 23 '12 at 5:12
    
@Will: the question mentions profinite groups at the end already. –  KConrad Oct 23 '12 at 5:53
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6 Answers 6

You may also read Chapter 13 of Kurosh's book. For instance, it contains a proof of Baer's theorem (cited by @Igor) which says that

all p-Sylow subgroups of a locally normal group are isomorphic.

Locally normal means periodic with finite conjugacy classes.

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A number of older papers by V.P. Platonov (in Russian, often followed by English translations) deal with periodic linear groups or linear algebraic groups in which the notions of Sylow theory make sense and where some results from the finite case actually generalize. One of the more substantial papers deals especially with conjugacy theorems:

The theory of algebraic linear groups and periodic groups. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 30 1966 573–620.

In other papers Platonov also works with classes of topological groups in a similar spirit.

P.S. Concerning sources, the long 1966 paper appears in an English translation (by the group theorist Kurt Hirsch) in volume 69 of the AMS Translations (Series 2), 1969; but this doesn't seem to be accessible online. There is a Google Scholar entry containing a full text PDF version of the Russian original here.

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Well, the wikipedia gives an example of a sylow theorem, and there is more on this in these notes by Igusa. There is also the following paper of Baer:

Sylow theorems for infinite groups

Reinhold Baer Source: Duke Math. J. Volume 6, Number 3 (1940), 598-614.

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Amalgams of finite groups provide another example. Let $A$ and $B$ be finite groups and let $C = A \cap B.$ Suppose that $P$ is a Sylow $p$-subgroup of $A$, and that $C$ contains a Sylow $p$-subgroup of $B.$ Then the amalgam $A*_{C}B$ has a unique conjugacy class of maximal finite $p$-subgroups, but is an infinite group as long as $C$ is proper in both $A$ and $B.$. In fact, the process an then iterated to the case where $A$ and $B$ may themselves be amalgams of finite groups of this type, and so on. For general results on amalgams, see J-P. Serre's book "Trees". For applications of this type of construction to fusion systems on finite $p$-groups, see two recent papers of mine in Journal of Algebra and Transactions of the AMS.

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The best reference for this subject is the book of martyn Dixon: Locally finite groups and Sylow theory.

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In groups of finite Morley rank there is a Sylow theory for the prime $p=2$.

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