Geodesic
Infinite groups containing a proper Hughes subgroup $H_3(G)$
Algebra i logika, Tome 60 (2021) no. 3, pp. 298-302.

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We consider groups in which $H_3(G)$ is both nontrivial and proper. In particular, it is proved that in such a group, $|G:H_3(G)|=3$.
Keywords: Hughes subgroup, Engel group, group of exponent $3$. 
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Wenbin Guo; D. V. Lytkina; V. D. Mazurov. Infinite groups containing a proper Hughes subgroup $H_3(G)$. Algebra i logika, Tome 60 (2021) no. 3, pp. 298-302. http://geodesic.mathdoc.fr/item/AL_2021_60_3_a2/

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