Notation : $\times $ used for Cartesian product, Sym means symmetric group, $\tau, \Delta \subseteq \Sigma$ and $\le$ for subgroup.
Input : $L\le Sym(\tau) \times Sym(\Delta)$; $z \in Sym(\tau \times \Delta);$ $\pi \subseteq \tau \times \Delta$.
Output : $(Lz)_{\pi} = \{x \in Lz \mid \pi ^x = \pi\}.$
Output is either $\phi$ or a right coset( of the group $L_{\pi}$).
I am not able to understand $(Lz)_{\pi}$ completely. $L$ makes sense to me it is a subgroup as given above. $\pi$ is also not a problem but $Lz$ I am not able to understand. As given in the problem statement $z \in Sym(\tau \times \Delta)$, but it should be that $z \in Sym(\tau )\times Sym( \Delta)$.
My questions : 1) Is $Sym(\tau )\times Sym( \Delta) \le Sym(\tau \times \Delta)$ up to isomorphism ?
2) How $(Lz)_{\pi}$ is a coset ?
The answer to Question 1 is yes. The action is given by $(a,b)^{(g,h)} = (a^g,b^h)$ for $(a,b) \in \tau \times \Sigma$ and $(g,h) \in {\rm Sym}(\tau) \times {\rm Sym}(\Sigma)$. Or, in other words, the element $(g,h) \in {\rm Sym}(\tau) \times {\rm Sym}(\Sigma)$ is mapped to the element $\phi(g,h) \in {\rm Sym}(\tau \times \Sigma)$ that maps $(a,b) \in \tau \times \Sigma$ to $(a^g,b^h)$.
For Question 2, suppose that $(Lz)_\pi$ is nonempty and hence $lz \in (Lz)_\pi$ for some $l \in L$. Then, for $l' \in L$, we have $$l'z \in (Lz)_\pi \Leftrightarrow \pi^{lz} = \pi^{l'z} \Leftrightarrow l'l^{-1} \in L_\pi,$$ so $(Lz)_\pi$ is the right coset $L_\pi lz$ of $L_\pi$.