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I define a symmetric group to be a group isomorphic to the group of all bijections on some set $S$. $S$ does not have to be finite, it could be infinite. My question is, is the class $K$ of all symmetric groups an elementary class? Also, bonus question, if it is not an elementary class, is $Th(K)$ finitely axiomatizable?

Shaun
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user107952
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    No, for instance the Lowenheim-Skolem Theorem would give a countable model, but there is no countable symmetric group. See also here for what is probably a duplicate. That said, the situation for finite symmetric groups is quite different, as they can be axiomatized (surprisingly, by a single sentence!). See here. – Chris Grossack Dec 07 '23 at 17:13

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This class is not elementary, because it does not have any countably infinite models.

For a bit more information on the theory of (finite) symmetric groups, see my answer here: https://math.stackexchange.com/a/4102001/7062

The sentence $\varphi$ given there is true of all symmetric groups, and its finite models are exactly the finite symmetric groups. But this does not establish that the theory of all symmetric groups is finitely axiomatizable - it's possible that there are additional sentences $\psi$ which are true of all symmetric groups but which are not entailed by $\varphi$ (because $\varphi$ may have infinite models which fail to satisfy $\psi$).

Shelah's paper First order theory of permutation groups may contain the answer to your bonus question, but I have not read it.

Alex Kruckman
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