Is there a name for "co-normal" subgroups? I have in mind a situation where a group $G$ has two subgroups $H$ and $K$ with the property that each of them is closed under conjugation by elements of the other.
More formally: for $h \in H$ and $k \in K$ then $h^{-1} k h \in K$ and $k^{-1}h k \in H$.
Is there a name for this?
 A: I would call them "mutually normalizing subgroups", and a Google search shows that other people have used this (which doesn't prove that there is not some more commonly used name).
A: $H$ is stable under conjugation by elements of $K$ if, and only if, $K$ is a subgroup of the normalizer $N_G(H)$ (which is the largest subgroup of $G$ which contains $H$ as a normal subgroup).
I don't think there is a name for the situation where $K\subset N_G(H)$ and $H\subset N_G(K)$.
A: A brief description is that $[H,K] \leq H \cap K$.
Suppose $G$ is a group with two subgroups $H,K$.
If $H$ normalizes $K$, then $k^{-1} \left( h^{-1} k h \right) \leq KK = K$ so $[H,K] \leq K$. Conversely if $[H,K] \leq K$, then $h^{-1} k h = k \cdot \left( k^{-1} h^{-1} k h \right) \leq KK = K$, so $H$ normalizes $K$.
Switching the roles of $H$ and $K$ gives that $K$ normalizes $H$ iff $[H,K] \leq H$.
Hence $H$ and $K$ normalize each other iff $[H,K] \leq H \cap K$.
A: By far the most popular according to Google is "$H$ and $K$ normalize each other".
A: Never seen this situation explored and don't think there's a name for it. Let's say that $(H,K)$ is a conormal pair if they satisfy the property you consider.
Two elementary observations are:
Every subgroup belongs to a conormal pair since $(K,\{e_G\})$ is conormal for every $K<G$. So, given $K$ one may ask about the (?) largest $H$ such that $(H,K)$ is conormal.
If $G=HK$ and $(H,K)$ is conormal, then both $H$ and $K$ are normal in $G$.
