# Has this connection between groups and the symmetries of relations been studied?

Given four distinct points $a,b,c,d$ on a circle, we can (speaking informally) choose in a unique way two pairs separating each other. For example the pairs $a,c$ and $b,d$ separate each other, if going around the circle starting from $a$, we pass the four points in the order $a\,b\,c\,d$ or (going the other way round) in the order $a\,d\,c\,b$. This is an informal description of a $4$-ary relation $S$ (in this case on the set of points of a circle), which is known in Geometry as the Separation relation.

Now $S$ is closely related to the group $D_8$ of symmetries of a square in the following sense: Given four elements $a_1,a_2,a_3,a_4$ (in this case points on our circle) and a permutation $\pi \in S_4$, we have $$S\,a_1a_2a_3a_4 \quad\Longrightarrow \quad ( S\,a_{\pi(1)}a_{\pi(2)} a_{\pi(3)}a_{\pi(4)} \, \Leftrightarrow \, \pi \in D_8)$$ (where $D_8$ is considered as subgroup of $S_4$).

My question: Has this sort of connection between a group $G$ and the symmetries of a relation $R$ been studied? Is there a term for this situation, e.g. “$R$ is $G$-symmetric“? I wouldn't want to make up a word if there is already one in usage.

## 1 Answer

I'm not too sure about existing research on symmetries of relations, but I can tell you how to formulate this in terms of group actions.

A $4$-ary relation $S$ is a subset of a Cartesian product $X_1\times X_2\times X_3\times X_4$. (In the case of your example, all the $X_i$ are the same.) As you've said, for $\pi \in S_4$ and $(a_1,a_2,a_3,a_4)\in X_1\times X_2\times X_3\times X_4$, we can define $$\pi \cdot (a_1,a_2,a_3,a_4) = (a_{\pi(1)},a_{\pi(2)},a_{\pi(3)},a_{\pi(4)})$$

For short, we say that $S_4$ acts on $X_1\times X_2\times X_3\times X_4$ by permutation.

The stabilizer in $S_4$ of an element $\overline{a}=(a_1,a_2,a_3,a_4)$ is the set $$\operatorname{Stab}_{S_4}(\overline{a}) = \{\pi \in S_4 : \pi \cdot \overline{a} \in S\}$$

So what you've stated is that $D_8=\operatorname{Stab}_{S_4}(\overline{a})$ for every $\overline{a}\in X$. (Though you should probably be careful here, as there are three different $D_8$ subgroups of $S_4$. You probably mean that they are isomorphic to $D_8$.) From here, you could look at the associated orbits in more detail, and that may give you some insight towards the structure of your relation.