# On the number $n_0$ in Shelah's construction of a Jonsson group

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:

The word $$r$$ in Fact 2.4 is here:

Just after Fact 2.4 in Remarks Shelah writes:

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 $$ax(ay)ax(ay)^2ax(ay)^3\cdots ax(ay)^{80}:$$

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?

• I don't think you included enough context to determine whether the word $r$ should start with $z^{-1}xa$ instead of $z^{-1}xz$. – YCor Oct 28 '18 at 18:25
• @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 $r$. – Taras Banakh Oct 28 '18 at 19:13