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Recall that a subgroup $H$ of a group $G$ is called a TI-subgroup if for every $g \in G$ it is $H \cap H^g \in \{1,H\}$. We say that a group $G$ is a DTI-group if the derived subgroups of all of its subgroups are TI-subgroups.

Question: Is it true that a non-trivial finite perfect DTI-group is necessarily simple?

Remark: I know that -- provided that it exists -- a non-trivial finite non-simple perfect DTI-group necessarily is minimal non-solvable.

Related question: References on a certain generalization of Dedekind groups

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  • $\begingroup$ I was wondering whether the non-split version of $2^3.GL_3(2)$ might be a potential candidate. do you have an easy way to test this? $\endgroup$
    – Nick Gill
    Commented May 13, 2020 at 16:02
  • $\begingroup$ @NickGill Running a check over the perfect groups of order less than about 50000 with GAP did not reveal a non-simple perfect DTI-group. -- So any counterexample (if it exists!) must be bigger. $\endgroup$
    – Stefan Kohl ♦
    Commented May 13, 2020 at 17:04

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