TITLE: Union of conjugates of a subgroup
QUESTION [6 upvotes]: Let $G$ be a finite group, $H \leq G$ a proper subgroup. It is well known that the union of the conjugates of $H$ does not cover $G$. I would like to know of more precise results (even in special cases) saying that the union of conjugates misses some structure.
For example, if $G$ is a Frobenius group, the complement of the union of conjugates is a subgroup (when $1$ is added). Is there any generalization of it? Maybe one could hope to find a submonoid (in the complement) in a more general case?
I would like to be able to deduce that this union misses something more specific than just some element
It is possible that there is something we can say if $G$ is a $p$-group.

REPLY [3 votes]: I'm not sure whether it is really relevant, but there is a generalization by Wielandt of Frobenius's theorem which may be interpreted as follows: Let $G$ be a transitive permutation group on a  set $\Omega$ , and let $G_{\alpha}$ be a point stabilizer. Suppose that $G_{\alpha} \neq \langle G_{\alpha} \cap G_{\beta} : \beta \neq \alpha \rangle.$ Then there is a proper normal subgroup of $G$ which is still transitive on $\Omega.$
This is a permutation group theoretic consequence  of Wielandt's generalization of Frobenius's theorem : if $H$ is a subgroup of a finite group $G$ and there is $K \lhd H$ with $H \cap H^{g} \leq K$ for all $g \in G \backslash H,$ then there is $L \lhd G$ with $G = HL$ and $L \cap H = K.$