Geodesic
A generalization of groups with many almost normal subgroups
Algebra and discrete mathematics, Tome 9 (2010) no. 1, pp. 79-85.

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A subgroup $H$ of a group $G$ is called almost normal in $G$ if it has finitely many conjugates in $G$. A classic result of B. H. Neumann informs us that $|G:\mathbf{Z}(G)|$ is finite if and only if each $H$ is almost normal in $G$. Starting from this result, we investigate the structure of a group in which each non-finitely generated subgroup satisfies a property, which is weaker to be almost normal.
Keywords: Dietzmann classes; anti-$\mathfrak{X}C$-groups; groups with $\mathfrak{X}$-classes of conjugate subgroups; Chernikov groups. 
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     author = {Francesco G. Russo},
     title = {A generalization of groups with many almost normal subgroups},
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Francesco G. Russo. A generalization of groups with many almost normal subgroups. Algebra and discrete mathematics, Tome 9 (2010) no. 1, pp. 79-85. http://geodesic.mathdoc.fr/item/ADM_2010_9_1_a6/