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Let $G$ be a group. It may be that $G$ has a subgroup $H$ that is the only one of the given isomorphism type, or at least contains all others of that isomorphism type. Somewhat weaker, it may be that $H$ contains all (sub)normal subgroups isomorphic to $H$. For instance, this is true of the Fitting subgroup of a finite group, as it is nilpotent and contains all nilpotent subnormal subgroups of $G$.

What I'm interested in is the opposite situation. What if, for every non-trivial subnormal subgroup $H$ of $G$, the subnormal subgroups of $G$ isomorphic to $H$ generate $G$? If $G$ is finite, then it's either characteristically simple or a $p$-group, I think (consider the generalised Fitting subgroup), and if it's a $p$-group then there's probably a lot more to be said. The question is also interesting in the context of residually finite groups because it has a natural link with questions about commensurators, especially if one restricts attention to subnormal subgroups of finite index. In particular, subnormal subgroups of finite index that are 'one of a kind' serve as obstacles to isomorphisms between subgroups of different finite indices.

Does anyone know of work done in this area (perhaps with more/stronger conditions)?

Edit: It occurs to me that in the case of finite p-groups, it is useful to know a subgroup is 'one of a kind' when looking at fusion systems, since such a subgroup is automatically weakly closed. What fusion systems are there with no proper non-trivial weakly closed subgroups?

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# Isomorphism types of subnormal subgroups

Let $G$ be a group. It may be that $G$ has a subgroup $H$ that is the only one of the given isomorphism type, or at least contains all others of that isomorphism type. Somewhat weaker, it may be that $H$ contains all (sub)normal subgroups isomorphic to $H$. For instance, this is true of the Fitting subgroup of a finite group, as it is nilpotent and contains all nilpotent subnormal subgroups of $G$.

What I'm interested in is the opposite situation. What if, for every non-trivial subnormal subgroup $H$ of $G$, the subnormal subgroups of $G$ isomorphic to $H$ generate $G$? If $G$ is finite, then it's either characteristically simple or a $p$-group, I think (consider the generalised Fitting subgroup), and if it's a $p$-group then there's probably a lot more to be said. The question is also interesting in the context of residually finite groups because it has a natural link with questions about commensurators, especially if one restricts attention to subnormal subgroups of finite index. In particular, subnormal subgroups of finite index that are 'one of a kind' serve as obstacles to isomorphisms between subgroups of different finite indices.

Does anyone know of work done in this area (perhaps with more/stronger conditions)?