**Notation**

- $\le$ is used for the subgroup relation;
- $P$ means polynomial time in input size;
- $\Omega = \{1,2,3,\cdots,n\}$ is a input domain;
- $\mathrm{Sym}(\Omega)$ means the symmetric group on $\Omega$;
- $G = \langle A \rangle $ means the subgroup $G$ generated by the subset $A$ of $\mathrm{Sym}(\Omega)$.

The *normal centralizer problem* is defined as follows:

**Given:** $G = \langle A \rangle, H = \langle B \rangle \le \text{Sym}(\Omega)$, where $G$ normalizes $H$.

**Find :** $C_G(H) = \{g \in G \mid gh =hg, \forall h \in H\}$

**Question :** Is this problem in $P$? Give a polynomial time algorithm if answer is yes. I know that if we drop the normal condition from the above problem then the new version will not be in $P$. Also note that computing the normalizer of a subgroup $H$ is in P.

Please note that I have asked the same question on theoretical computer science exchange (link) a month back but did not get any response.