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I have two questions:

  1. In a text, I read that a group permutes pairs of faces of a solid transitively. Geometrically, what are they referring to, and what is an example of when a group may not permute some aspect of a geometric object transitively?
  2. The definition I usually find for transitive is that there is only one orbit, or that $Gx=Gy$ for all $x,y$. How does this relate to the usual notion of transitivity (like, in equivalence relations?)
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up vote 7 down vote accepted

There is no direct relation between the uses of "transitive" in "transitive relation" and in "transitive action". The relation of being in the same orbit is transitive, but that is true for any group action.

However both senses are derived from the latin verb "transire", meaning something like "going across" or "going through". A transitive action allows you to go across from any point to any other point, and a transitive relation allows you to go through any chain of relations to relate the outer terms.

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