Is it known whether the Schur Multiplier of the Tarski monsters are finitely generated?
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5$\begingroup$ Can you flesh this out just a little? $\endgroup$– Jon BannonCommented Oct 2, 2015 at 12:13
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2$\begingroup$ And include a precise definition for "Tarski monster", It sometimes refers to non-locally-finite quasi-finite group (quasi-finite means: has no infinite proper subgroup), and sometimes to some particular instances of those (e.g., assuming every proper subgroup is cyclic, or even cyclic of fixed order), or also to refer to some specific constructions of such groups. $\endgroup$– YCorCommented Oct 2, 2015 at 12:31
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$\begingroup$ I am looking at groupprops.subwiki.org/wiki/Tarski_group. I have asked a question here if schur multiplier of Noetherian groups is finitely generated. Tarski Monster is an example of Noetherian group, so hence the above question. $\endgroup$– user114539Commented Oct 2, 2015 at 15:19
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$\begingroup$ The subwiki link being possibly periodically modified, let me copy its definition (so that the definition makes sense): it's an infinite group which, for some prime $p$, has all its proper subgroups cyclic of order $p$. $\endgroup$– YCorCommented Apr 8, 2018 at 18:52
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There is no such thing as "the Tarski monster". There exists an uncountable set of Tarski monsters, as was first proved by Olshanskii. His Tarski monsters have infinitely generated Schur multipliers as shown in Section 5 of "Lacunary hyperbolic groups". See also the earlier paper by Ashmanov and Olshanskii MR0829100 where a similar statement was proved for the free Burnside groups (the proof is essentially the same).