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I think the parenthetical remark in Definition 2.4 here is wrong if $X$ is empty.

If $X$ is a $G$-set and if $G · x = X$ for one (or equivalently all) $x ∈ X$, we say that $G$ acts transitively on $X$.

Why I think this is wrong:

For the first direction of one $x$ implies all $x$: The existence of an $x \in X$ implies $X$ is not empty. No problem here.

For the other direction: We could have "all $x \in X$" as holding vacuously for empty $X$. Therefore, no such $x \in X$ exists.

Do I misunderstand?

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    You're right. If you really want to consider action on the empty set, the definition should go with 'all'.. – Berci Oct 14 '19 at 14:23
  • Merci, @Berci. You can answer if you want. –  Oct 14 '19 at 14:27
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    Just some suggestions on using this site: If you're referring to a small quote from a document, then rather than linking to a pdf, it's better form to copy out the quote (in this case Definition 2.4) and give a citation to where you found it. pdfs can change and become inaccessible. Also, if you think something is wrong, it's better form to explain why you think it's wrong, rather than just asking for confirmation. – Alex Kruckman Oct 14 '19 at 17:37
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    I would disagree with Berci: the more useful definition is "for one", not "for all". That is, the action on the empty set should not be considered transitive. – Eric Wofsey Oct 14 '19 at 17:40
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    Hmm.. Possible.. Why not? – Berci Oct 14 '19 at 23:46
  • @Berci You have to tag using the "@" –  Oct 16 '19 at 03:40
  • @EricWofsey Berci says "Hmm.. Possible.. Why not?" –  Oct 16 '19 at 03:40
  • @AlexKruckman Great suggestions. Why not make your suggestions into rules? I edited the post. Thanks. –  Oct 16 '19 at 03:41
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    @Berci: It's the same reason that $1$ is not considered prime. It's convenient for transitive actions to be those with exactly one orbit, so that every group action decomposes uniquely as a disjoint union of transitive actions. – Eric Wofsey Oct 16 '19 at 03:52
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    $X$ is a $G-$ set, and a mapping is defined in general on a non-empty set, then how can $X$ be empty? –  Oct 16 '19 at 04:04
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    There is one mapping from the empty set. – Berci Oct 16 '19 at 06:40
  • @Berci Again, You have to tag using the "@" –  Oct 16 '19 at 08:17
  • @EricWofsey Berci says "There is one mapping from the empty set." –  Oct 16 '19 at 08:17
  • @who You can certainly define a $G$-action on the empty set! You can equivalently consider this as the unique function $G \times \varnothing \to \varnothing$ (which will automatically be associative and unital) or as the unique group homomorphism $G \to Aut(\varnothing)$. – Aaron Mazel-Gee Nov 15 '20 at 16:36

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