Locally soluble groups with all nontrivial normal subgroups isomorphic
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Let $G$ be an infinite, locally soluble group which is isomorphic to all its nontrivial normal subgroups. If $G/G'$ has finite $p$-rank for $p=0$ and for all primes $p$, then $G$ is cyclic.
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Lennox, J. C. et al. “Locally soluble groups with all nontrivial normal subgroups isomorphic”. Publicacions Matemàtiques, vol.VOL 38, no. 1, pp. 203-5, https://raco.cat/index.php/PublicacionsMatematiques/article/view/37800.
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