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I am looking for books, papers, or even webpages, that have collected many counter examples in group theory (which, I guess, are just examples in group theory).

I am particularly interested in sources with a focus on infinite, finitely generated, groups. As some examples, a reference may contain things like:

  • Tarski monster groups: an infinite groups where every element has the same finite order, non-amenable group that does not contain a free subgroup of rank two etc

  • Baumslag-Solitar groups give examples of finitely presented (one-relator) non-Hopfian groups, etc

  • Groups of intermediate growth, they are non-elementarily amenable and they have intermediate growth.

  • Infinite groups without a non-discrete Hausdorff group topology

  • Fintely presented groups with strange properties (where maybe infinitely presented are easier to find/construct)

  • Finitely presented groups without solvable word problem.

  • Counter examples restricted to groups that are considered nicer than arbitrary finitely generated group (finitely presented groups), like hyperbolic groups or automatic groups.

  • ....

For online references it would be especially cool if you could search for groups with certain properties in a similar vain to the $\pi$-base website. Although I would really like such a site to contain references to the literature. (I don't think such a site exists, as I would have found it but, hopefully I am mistaken)

Please one source per answer, and if possible summarize what it covers. I know it it is basically hopeless for one central source, so I made this a big list. Plus this opens it up to references that are not focused on infinite, finitely presented sources which should hopefully make this more useful to others.

A sort of relevent question, on mathoverflow, is Counterexamples in Algebra?

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  • $\begingroup$ I am not sure, maybe this question would be better off asking for individual groups that are counter examples to a lot of things... $\endgroup$ – Paul Plummer Apr 29 '15 at 2:50

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