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.



Union of conjugates of a subgroup
To avoid examples like subgroups of cyclic groups, we could assume that $H$ is core-free in $G$. Then the question is equivalent to asking about fixed point free elements of transitive permutation groups.


In a transitive $p$-group, a minimal normal subgroup has order $p$ and must be fixed-point-free (since the stabilizer is core-free), so the complement (plus identity) contains a subgroup of order $p$.


In general, there need be no fixed-point-free elements of prime order. For example, TransitiveGroup(12,47) (in the GAP/Magma database) is a group of order $72$ with stabilizer dihedral of order $6$. There are $54$ fixed-point-free elements, all of order $4$, so the complement plus identity contains no submonoid.


