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In the paper "On a Problem of Kurosh, Jonsson Groups, and Applications", Shelah proved the following

Theorem 2.1. There exists a number $n_0$ such that for every infinite cardinal $\lambda$ with $\lambda^+=2^\lambda$ there exists a group $G$ of cardinality $|G|=\lambda^+$ such that $G=A^{n_0}$ for any subset $A\subset G$ of cardinality $|A|=|G|$.

I would like to deduce the value of $n_0$ from the Shelah's proof (of his Theorem 2.1).

On page 383 Shelah writes that $n_0$ is the length of the word $r$ in Fact 2.4:


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The word $r$ in Fact 2.4 is here:

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Just after Fact 2.4 in Remarks Shelah writes:

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  1. I have a doubt that this word $r$ should start with $z^{-1}xz$ and suggest that it should be $z^{-1}xa$.

Note that in Remarks, Shelah writes that $r$ consists of instances of $xa$ and $ya$, then multiplied by $z^{-1}$. So, it is natural to suggest that only one letter $z$ should occur at the very end (or beginning, depending from which side start counting) of the word $r$.

Also discussing the Small Cancellation Theory in $\S 1$, Shelah cites a result of Schupp concerning the symmetrized closure of the word $$ax(ay)ax(ay)^2ax(ay)^3\cdots ax(ay)^{80}:$$

enter image description here


  1. The length of the word $r$ in Fact 2.4 equals $1+2\cdot 80+2\frac{80\cdot 81}2=1+80\cdot 83=6641$.

  2. It seems that the Shelah's construction for any subset $A\subset G$ of cardinality $|A|=|G|=\lambda^+$ and any element $z\in G$ yields elements $x,a,b\in A$ such that $r(a,x,y,z)=1$, so if my suggestion 1 is true, then $z$ belongs to $A^{l(r)-1}=A^{6440}$, where $l(r)=6641$ is the length of the word $r$.

Therefore, if my suggestions (1)-(3) are correct, then Theorem 2.1 has the following more precise form:

Theorem. For every infinite cardinal $\lambda$ with $\lambda^+=2^\lambda$ there exists a group $G$ of cardinality $|G|=\lambda^+$ such that $G=A^{6440}$ for any subset $A\subset G$ of cardinality $|A|=|G|$.

So, my question:

Is $n_0$ in Shelah's Theorem 2.1 equal to 6640 or 6641? Or maybe to some other number?

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  • $\begingroup$ I don't think you included enough context to determine whether the word $r$ should start with $z^{-1}xa$ instead of $z^{-1}xz$. $\endgroup$ – YCor Oct 28 '18 at 18:25
  • $\begingroup$ @YCor I added the link to the original paper of Shelah and also copied Remark following Fact 2.4 in which Shelah explains why he has chosen this particular word $r$. $\endgroup$ – Taras Banakh Oct 28 '18 at 19:13

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