# Making use of partial knowledge of $G \cap H$

We can prove some parts of our search don't have to be performed. Early in search we find the permutation $(4,5) \in G \cap H$. Later, we try finding a permutation where $1^p = 4$. There is no need after this to search for a permutation where $1^p = 5$. Why? There are two cases:

** There is a permutation where $1^p = 4$ (in this problem there is, $(1,4)(2,6)(3,5) \in G \cap H$). We can find a permutation where $1^p = 5$ by multiplying these two permutations together, to get $(1,5,3,4)(2,6)$.

** If we had found no permutation where $1^p = 4$ in $G \cap H$, there is no point searching for one where $1^p=5$, as if we found such a permutation, we could again multiply it by $(4,5)$ and get a permutation where $1^p = 4$, which can't exist!

How can we formalise this idea? As we find members of $G \cap H$, we keep track of the orbits of $G \cap H$ and only check one value in each orbit. An exact discussion of this algorithm will appear in a future post!

# Proving parts of search contain no element of $G \cap H$.

This is where the majority of the research into improving backtrack search in permutation groups (and backtrack searches in general) is performed.

Looking at the output of FindExtendingElement, there are a number of easy ways it can be improved. We certainly shouldn't test mappings like [ 3, 2, 1, 1 ], as permutations are invertible!

More generally, let us consider the second stage of our search, when we are looking for $G_1 \cap H_1$. We can ask GAP to give us the orbits of $G_1$ and $H_1$:

Orbits(Stabilizer(G, 1));
# [ [ 2, 3 ], [ 4, 5, 6 ] ]
Orbits(Stabilizer(H, 1));
# [ [ 2, 4, 5 ], [ 3, 6 ] ]


From this, we can deduce the orbit of $2$ in $G_1 \cap H_1$ must be contained in ${2,3} \cap {2,4,5}$, which means that actually the orbit of $2$ is just ${2}$!

By the same reasoning, both $3$ and $6$ are fixed, leaving the only non-trivial orbit as ${4,5}$. Of course, this does not mean that $G_1 \cap H_1$ contains a permutation $p$ where $4^p = 5$, this orbit may still split. However, this is the only case which we need to consider. We have reduced our search for $G_1 \cap H_1$ to having to consider a single permutation: $(4,5)$!