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There are two groups, $G_1$ and $G_2$. They are both acting on a set $S$.

$S$ may have some structure. The groups may too. The actions respect them.

$G_1$ is mysterious. Perhaps all we know about it is the way it acts on $S$. We'd like to know more.

$G_2$ is well-known. Its structure and action are totally transparent.

We might be able to learn about $G_1$ from watching how its action interacts with the action of $G_2$. Intersection of their orbits, for instance.

Is this situation systematically studied under some name? Beyond the case when the actions commute.

There are two groups, $G_1$ and $G_2$. They are both acting on a set $S$.

$S$ may have some structure. The groups may too. The actions respect them.

$G_1$ is mysterious. Perhaps all we know about it is the way it acts on $S$. We'd like to know more.

$G_2$ is well-known.

We might be able to learn about $G_1$ from watching how its action interacts with the action of $G_2$. Intersection of their orbits, for instance.

Is this situation systematically studied under some name? Beyond the case when the actions commute.

There are two groups, $G_1$ and $G_2$. They are both acting on a set $S$.

$S$ may have some structure. The groups may too. The actions respect it.

$G_1$ is mysterious. Perhaps all we know about it is the way it acts on $S$. We'd like to know more.

$G_2$ is well-known. Its structure and its $S$-action are totally transparent.

We might be able to learn about $G_1$ from watching how its action interacts with the action of $G_2$. Interplay of their orbits, for instance, might tell us about subgroups.

Is this situation systematically studied under some name? Beyond the case when the actions commute.

There are two groups, $G_1$ and $G_2$. They are both acting on a set $S$.

$S$ may have some structure. The groups may too. The actions respect them.

$G_1$ is mysterious. Perhaps all we know about it is the way it acts on $S$. We'd like to know more.

$G_2$ is well-known. Its structure and action are totally transparent.

We might be able to learn about $G_1$ from watching how its action interacts with the action of $G_2$. Intersection of their orbits, for instance.

Is this situation systematically studied under some name? Beyond the case when the actions commute.

There are two groups, $G_1$ and $G_2$. They are both acting on a set $S$.

$S$ may have some structure. The groups may too. The actions respect it.

$G_1$ is mysterious. Perhaps all we know about it is the way it acts on $S$. We'd like to know more.

$G_2$ is well-known. Its structure and its $S$-action are totally transparent.

We might be able to learn about $G_1$ from watching how its action interacts with the action of $G_2$. For instance, how their orbits overlap and intersect might tell about subgroups.

Is this sort of thing systematically studied under some name? Beyond the case when the actions commute.

There are two groups, $G_1$ and $G_2$. They are both acting on a set $S$.

$S$ may have some structure. The groups may too. The actions respect it.

$G_1$ is mysterious. Perhaps all we know about it is the way it acts on $S$. We'd like to know more.

$G_2$ is well-known. Its structure and its $S$-action are totally transparent.

We might be able to learn about $G_1$ from watching how its action interacts with the action of $G_2$. Interplay of their orbits, for instance, might tell us about subgroups.

Is this situation systematically studied under some name? Beyond the case when the actions commute.