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Recall that a group is called a Dedekind group if all of its subgroups are normal. Also recall that a weaker property of a subgroup than normality is that of being a TI-subgroup: a subgroup $H$ of a group $G$ is called a TI-subgroup if for every $g \in G$ it is $H \cap H^g \in \{1,H\}$.

The groups all of whose subgroups are TI-subgroups have been classified in:

G. Walls: Trivial intersection groups, Arch. Math. 32, 1-4 (1979).

Now it is natural to weaken the condition to a certain extent, and to consider groups a certain subset of whose subgroups are TI-subgroups. -- So for example the groups all of whose abelian subgroups are TI-subgroups are classified in:

Guo, Xiuyun; Li, Shirong; Flavell, Paul: Finite groups whose Abelian subgroups are TI-subgroups, J. Algebra 307, No. 2, 565-569 (2007).

Question: Is anyone aware of any work having been done so far on groups the derived subgroups of all of whose subgroups are TI-subgroups?

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  • $\begingroup$ What is your motivation to choose subgroups of derived subgroup? $\endgroup$
    – Alireza Abdollahi
    Commented Aug 31, 2018 at 18:03
  • $\begingroup$ @AlirezaAbdollahi: Not subgroups of the derived subgroup -- but subgroups which occur as derived subgroups of subgroups (as written in the question). $\endgroup$
    – Stefan Kohl ♦
    Commented Aug 31, 2018 at 20:20
  • $\begingroup$ @AlirezaAbdollahi: My collaborator Leyli Jafari suggested the topic, and the aim of the question is to find out whether anyone else has already done some work on these groups, in order to avoid duplicating existing work. $\endgroup$
    – Stefan Kohl ♦
    Commented Aug 31, 2018 at 20:39
  • $\begingroup$ @StefaKohl I misunderstood the question. Anyway, interesting to know why she did not choose another subgroup instead of the derived subgroup? $\endgroup$
    – Alireza Abdollahi
    Commented Sep 2, 2018 at 8:56

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