I noted that the Dummit-Foote text often (not always though) skipped the condition of non-emptyness in defining certain terms e.g. in group the condition that the base set needs to be nonempty isn't taken into consideration and so for the set the $A$ on which a group act in course of defining group action. However in defining the set of all permutations on a set $\Omega,$ it's taken to be nonempty. Please help me to get why the authors follow such convention which goes against (I don't if it really is) the standard definitions of those terms.

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    $\begingroup$ The group axioms require the existence of an identity element, thus guaranteeing non-emptiness. $\endgroup$ – TonyK Aug 31 '13 at 14:39
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    $\begingroup$ For the group itself, non-emptiness follows from the existence of a unit. Every group acts (albeit in a rather uninteresting way) on the empty set, so there's no need to require the set the group acts on to be non-empty. $\endgroup$ – Daniel Fischer Aug 31 '13 at 14:39
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    $\begingroup$ $\exists e$ means there really exists an $e$. And in the $\forall g$, the universe over which the universal quantification roams includes at least $e$. $\endgroup$ – André Nicolas Aug 31 '13 at 14:57
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    $\begingroup$ Sounds like a cool book name: "Theory of empty groups". $\endgroup$ – rfauffar Aug 31 '13 at 18:25
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    $\begingroup$ @rfauffar Sounds like a short book. Perhaps with no pages... $\endgroup$ – rschwieb Sep 26 '17 at 13:20

The group axioms require the existence of an identity element, which guarantees non-emptiness.


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