I have encountered a mysterious condition on finite groups in my research, and would like help understanding it better.
Let $G$ be a finite group, and let $H\leq K\leq G$ be a chain of subgroup inclusions. I am interested in whether the following condition holds:
There exists $L\leq G$ such that for every conjugate $L^g$ of $L$ in $G$, $K\cap L^g$ is conjugate to $H$ in $K$.
In particular, I'm interested in
- necessary and/or sufficient conditions on $H$, $K$, and $G$ for this to hold, and
- necessary and/or sufficient conditions on $G$ for this to hold for all $H\leq K\leq G$.
For instance, a sufficient condition is $H$ being normal in $G$, since in that case we can take $L=H$, and the condition is satisfied. Therefore, if $G$ is Dedekind (i.e., all subgroups are normal), this condition is satisfied for all $H\leq K\leq G$. This is a rather restrictive condition, though, and there seem to be other groups (e.g., dihedral groups $D_p$ of order $2p$) for which this holds.
I have run some experiments in Magma to find all groups $G$ of order less than 256 for which this condition holds for all $H\leq K\leq G$. Some properties shared by all these groups include:
- They have derived length at most 2, or equivalently, are metabelian (i.e., an extension of an abelian group by an abelian group).
- Their Sylow subgroups are Dedekind. Explicitly, this means their Sylow 2-subgroups are either abelian or isomorphic to $Q_8\times C_2^r$, and for $p$ odd, their Sylow $p$-subgroups are abelian.
While these two conditions may be necessary, they are not sufficient. A counterexample is $G=D_{45}$, which has all Sylow subgroups cyclic (and is hence metabelian), but fails to satisfy the condition for the subgroups $C_2\leq D_3\leq D_{45}$, among others.
Any help in understanding this situation would be much appreciated! Thanks!
