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While reading "Chain conditions, elementary amenable groups, and descriptive set theory" by Phillip Wesolek and Jay William I stumbled upon the following statement in the proof of Lemma $4.8$:

Given a sequence of (countable) groups $\{A_i\}_{i \in \mathbb{N}}$, there exists a $2$-generated group $A$ which contains every $A_i$ and every proper subgroup of $A$ is infinite cyclic, infinite dihedral or contained in a conjugate of some $A_i$.

This is supposed to be a consequence of Theorem $2$ in "Efficient embeddings of countable groups" by A. Y. Olshanskii (Moscow Univ. Math. Bull. 44 (1989), no. 2,39–49) and even though I found the paper on mathnet.ru, it turned out to be in Russian which I cannot comprehend. I was unable to find a translation.

It would be great if anyone knows another source where I can find this construction (or one that fits the statement) or if someone knows the construction and is willing to explain it to me.

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    $\begingroup$ This is not just a consequence of Theorem 2, it is (part of) the exact statement of Theorem 2. Copying and pasting the theorem into any online translating tool will be more than sufficient to understand the meaning. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented May 19, 2021 at 8:59
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    $\begingroup$ The English translation of Ol'shanskii's paper is here: Economical embeddings of countable groups. (English. Russian original) Zbl 0711.20017 Mosc. Univ. Math. Bull. 44, No. 2, 39-49 (1989); translation from Vestn. Mosk. Univ., Ser. I 1989, No. 2, 28-34 (1989). $\endgroup$
    – Ashot Minasyan
    Commented May 19, 2021 at 16:45
  • $\begingroup$ @Carl-FredrikNybergBrodda : Thank you for pointing that out, I will try doing that and hope I will understand it. $\endgroup$
    – Matthias Klupsch
    Commented May 19, 2021 at 18:07
  • $\begingroup$ @AshotMinasyan : Thank you for the comment, I was able to find the same reference you cite. However, I could not find the translation itself, the issues of Mosc. Univ. Math. Bull. I could find online were more recent than the one from 1989 I am looking for. $\endgroup$
    – Matthias Klupsch
    Commented May 19, 2021 at 18:11

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