In recent work Lifting $N_\infty$ operads from conjugacy data on homotopical combinatorics / $N_\infty$ operads in equivariant homotopy theory, collaborators Scott Balchin, Ethan MacBrough, and I identified a class of groups we call lossless. Here is our definition:

A group $G$ is lossless when for all $K\le H\le G$ and $g\in G$ such that $gKg^{-1}\le H$, there exists $h\in N_G(H)$ (the normalizer of $H$ in $G$) such that $hKh^{-1}\le H$.

Here are our questions:

Is this class of groups (a) studied elsewhere in the literature (presumably under a different name)? or (b) specified by some more familiar string of conditions/adjectives?

For our purposes, we would be happy to restrict to $G$ finite.

We already know (see §2.2 of the paper) that finite solvable T-groups, $p$-groups of order at most $p^3$, and groups with cyclic normal subgroups of prime index are lossless. The group $\operatorname{SL}_2(\mathbb{F}_7)$ is lossy (i.e. not lossless) and we give a class of $p$-groups of order $p^4$ which are lossy in Example 2.18.

In recent work Lifting $N_\infty$ operads from conjugacy data on homotopical combinatorics / $N_\infty$ operads in equivariant homotopy theory, collaborators Scott Balchin, Ethan MacBrough, and I identified a class of groups we call lossless. Here is our definition:

A group $G$ is lossless when for all $K\le H\le G$ and $g\in G$ such that $gKg^{-1}\le H$, there exists $h\in N_G(H)$ (the normalizer of $H$ in $G$) such that $hKh^{-1} = gKg^{-1}$.

Here are our questions:

Is this class of groups (a) studied elsewhere in the literature (presumably under a different name)? or (b) specified by some more familiar string of conditions/adjectives?

For our purposes, we would be happy to restrict to $G$ finite.

We already know (see §2.2 of the paper) that finite solvable T-groups, $p$-groups of order at most $p^3$, and groups with cyclic normal subgroups of prime index are lossless. The group $\operatorname{SL}_2(\mathbb{F}_7)$ is lossy (i.e. not lossless) and we give a class of $p$-groups of order $p^4$ which are lossy in Example 2.18.

In recent work on homotopical combinatorics / $N_\infty$ operads in equivariant homotopy theory, collaborators and I identified a class of groups we call lossless. Here is our definition:

A group $G$ is lossless when for all $K\le H\le G$ and $g\in G$ such that $gKg^{-1}\le H$, there exists $h\in N_G(H)$ (the normalizer of $H$ in $G$) such that $hKh^{-1}\le H$.

Here are our questions:

Is this class of groups (a) studied elsewhere in the literature (presumably under a different name)? or (b) specified by some more familiar string of conditions/adjectives?

For our purposes, we would be happy to restrict to $G$ finite.

We already know (see $\S$2.2 of the paper) that finite solvable T-groups, $p$-groups of order at most $p^3$, and groups with cyclic normal subgroups of prime index are lossless. The group $\operatorname{SL}_2(\mathbb{F}_7)$ is lossy (i.e. not lossless) and we give a class of $p$-groups of order $p^4$ which are lossy in Example 2.18.

In recent work Lifting $N_\infty$ operads from conjugacy data on homotopical combinatorics / $N_\infty$ operads in equivariant homotopy theory, collaborators Scott Balchin, Ethan MacBrough, and I identified a class of groups we call lossless. Here is our definition:

A group $G$ is lossless when for all $K\le H\le G$ and $g\in G$ such that $gKg^{-1}\le H$, there exists $h\in N_G(H)$ (the normalizer of $H$ in $G$) such that $hKh^{-1}\le H$.

Here are our questions:

Is this class of groups (a) studied elsewhere in the literature (presumably under a different name)? or (b) specified by some more familiar string of conditions/adjectives?

For our purposes, we would be happy to restrict to $G$ finite.

We already know (see §2.2 of the paper) that finite solvable T-groups, $p$-groups of order at most $p^3$, and groups with cyclic normal subgroups of prime index are lossless. The group $\operatorname{SL}_2(\mathbb{F}_7)$ is lossy (i.e. not lossless) and we give a class of $p$-groups of order $p^4$ which are lossy in Example 2.18.