If $G$ is a finite $p$-group and $H$ is a proper subgroup of $G$, then, it is well known that the union of conjugates of $H$ in $G$ is proper subset of $G$. The problem I considering is the following:


**Question** Let $G$ be a finite $p$-group, $H$ be a non-normal subgroup, such that its normalizer $N\_G(H)$ is normal in $G$ (hence, all conjugates of $H$ are contained in $N\_G(H)$). Is the union of all conjugates of $H$, a proper subset of $N\_G(H)$?



Union of Conjugate Subsgroups in $p$-Groups
I believe the answer is yes, the union of the conjugates of $H$ is a proper subset of $N\_G(H)$.


Let $H < G$ be a counterexample with $|G|$ minimal. Since $N\_G(H) \lhd G$, the intersection $Z := N\_G(H) \cap Z(G)$ of $N\_G(H)$ with the centre of $G$ is nontrivial.


If $Z < H$, then $H/Z < G/Z$ is a smaller counterexample, whereas if $Z \not< H$ then there exists $g \in Z \setminus H$ and then $g$ is not contained in any conjugate of $H$.


