In recent work Lifting N∞ operads from conjugacy data on homotopical combinatorics / N∞ 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≤H≤G and g∈G such that gKg−1≤H, there exists h∈NG(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 p3, and groups with cyclic normal subgroups of prime index are lossless. The group SL2(F7) is lossy (i.e. not lossless) and we give a class of p-groups of order p4 which are lossy in Example 2.18.