Geodesic
An extension of a theorem of Neumann
Matematičeskie trudy, Tome 26 (2023) no. 1, pp. 41-46 
V. G. Durnev; A. I. Zetkina. An extension of a theorem of Neumann. Matematičeskie trudy, Tome 26 (2023) no. 1, pp. 41-46. http://geodesic.mathdoc.fr/item/MT_2023_26_1_a2/
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     author = {V. G. Durnev and A. I. Zetkina},
     title = {An extension of a theorem of {Neumann}},
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Voir la notice de l'article provenant de la source Math-Net.Ru

In the present article, we prove that every countable infinite group $G$ is embeddable into a countable infinite simple group $\overline{G}$ such that every equation of the form $$w(x_1,\dots,x_n)=g,$$ is solvable in $\overline{G}$, where $w$ is a nontrivial reduced group word in variables $x_1,\dots,x_n$, and $g\in G$.

[1] M. I. Kargalolov, Yu. I. Merzlyakov, Osnovy teorii grupp, Nauka, M., 1982

[2] R. Lindon, P. Shupp, Kombinatornaya teoriya grupp, Mir, M., 1980

[3] V. Magnus, A. Karras, D. Soliter, Kombinatornaya teoriya grupp. Predstavlenie grupp v terminakh obrazuyuschikh i sootnosheii, Nauka, M., 1974

[4] B. H. Neumann, “Adjunction of elements to groups”, J. Lond. Math. Soc., 18 (1943), 4–11