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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    $\begingroup$ @Colin Reid: You do not seem to be really talking about "isomorphism types" so much as "relative/intrinsic properties" (e.g., "is nilpotent and subnormal"). $\endgroup$ – Arturo Magidin Nov 22 '10 at 19:51
  • $\begingroup$ Can you give an example of a finite example which is not characteristically simple or of exponent $p$? $\endgroup$ – Jonathan Kiehlmann Nov 22 '10 at 19:52
  • $\begingroup$ @Arturo Magidin: I could ask about all subgroups instead, but for the application to commensurators it's enough to know about subnormal subgroups, which I'm hoping will restrict the possibilities more. (The same property without the word 'subnormal' is potentially weaker.) Isomorphism type is of course the strongest possible intrinsic property. $\endgroup$ – Colin Reid Nov 22 '10 at 21:13
  • $\begingroup$ @Jonny: Certainly any finite p-group example is generated by elements of order p (hence so are all its images). So any image of the group that is a regular p-group (eg $G/\gamma_{p-1}(G)$) would have exponent p. This probably isn't very helpful if p=2 but maybe it is for larger primes. $\endgroup$ – Colin Reid Nov 23 '10 at 12:49
  • $\begingroup$ @Jonathan Kiehlmann: The smallest examples are SmallGroup(32,49) and 32,50; (D8 x 2):2 and (Q8 x 2):2. They have the weird property (generated by each iso-class of subnormal subgroups), have exponent 4, and are not characteristically simple. $\endgroup$ – Jack Schmidt Nov 23 '10 at 16:14

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