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Is there a first-order theory of symmetric groups or group actions?

I'm curious whether there's a first-order axiomatization of the theory of a single symmetric group or group action, ideally one with finitely many axioms.

I'm ideally hoping for a constructive answer. (Collecting all the sentences that are true in every symmetric group might work, but as a theory it is not easy to use)

The following theory is my attempt to create a first-order theory for a group $G$ acting on a set $X$ such that $G$ is isomorphic to the automorphism group of $X$.

It works for finite groups, but breaks down for infinite groups. (Permutations of $\mathbb{N}$ with cofinitely many fixed points satisfy this theory but are not symmetric groups).

Our signature is algebraic and contains the following functions with the following types.

$$ \cdot : G \times G \to G \\ \square^{-1} : G \to G \\ \triangleright: G \times X \to X \\ 1 : G$$

Group axioms.

The only choice I made here that's a little weird is insisting on the existence of a right identity and then including another axiom saying that the right identity is also a left identity.

$$ \forall a b c : G \ldotp (a \cdot b) \cdot c = a \cdot (b \cdot c)$$ $$ \forall e : G \ldotp (\forall a : G \ldotp a \cdot e = a) \leftrightarrow e = 1 $$ $$ \forall e : G \ldotp 1 \cdot a = a $$ $$ \forall a : G \ldotp a \cdot a^{-1} = 1 \land a^{-1} \cdot a = 1 $$

Group action axioms

$$ \forall gh : G \ldotp \forall x : X \ldotp (g \cdot h) \triangleright x = g \triangleright h \triangleright x $$ $$ \forall x : X \ldotp 1 \triangleright x = x $$

Next I have an axiom saying that the group action is faithful.

$$ \forall g : G \ldotp (\forall x : X \ldotp g \cdot x = x) \leftrightarrow g = 1 $$

I have the following extensionality-like theorem:

$$ \text{Thm:}\;\;\; (\forall g h : G \ldotp (\forall x : X \ldotp g \triangleright x = h \triangleright x) \leftrightarrow g = h) $$

As proof, if $g = h$, then they act the same way on $X$. If $g \neq h$ but they act the same way on $X$, then $gh^{-1}$ fixes $X$ pointwise. However, $gh^{-1}$ is not the identity element of $G$, which is a contradiction.

Finally, as an axiom, I insist that on the existence of all transpositions.

$$ \forall x y : X \ldotp \exists! g: G \ldotp (g \triangleright x = y \land g \triangleright y = x) \land (\forall z : X \ldotp z \neq x \land z \neq y \to g \triangleright z = z) $$

For finite groups and sets, this construction works.

Any permutation over a finite set is a composition of finitely many transpositions. However, the set of permutations of $\mathbb{N}$ with cofinitely many fixed points satisfies this theory, but isn't a symmetric group.

Greg Nisbet
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    By the downward Lowenheim-Skolem theorem, there will be a countable group $G\subseteq S_\mathbb{N}$ such that $(G,\mathbb{N})\preccurlyeq(S_\mathbb{N},\mathbb{N})$ no matter how we set up a group $A$ acting on a set $X$ as a structure $(A,X)$. – Noah Schweber Sep 08 '21 at 01:18
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    That said, in the finite realm there is a surprising (to me at least) positive result - see here. – Noah Schweber Sep 08 '21 at 01:24
  • @NoahSchweber ... Right. Thanks. If you could do this for infinite symmetric groups, then you'd have a first-order theory that's able to control its own (infinite) cardinality. – Greg Nisbet Sep 08 '21 at 01:25

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Turning my comments into an answer:

The downward Lowenheim-Skolem theorem rules out anything like a positive answer: given any reasonable way to associate a structure $\mathcal{M}_{G,X}$ to a set $X$ and a group of permutations $G\subseteq Sym_X$, there will be an infinite $X$ and a "small" subgroup (in particular, proper) $G\subsetneq Sym_X$ with $\mathcal{M}_{G,X}\preccurlyeq\mathcal{M}_{Sym_X,X}$.

On the other hand, as often happens the finite model theory situation is quite different: there is a single first-order sentence $\sigma$ such that if $G$ is a finite group then $G\models\sigma$ iff $G\cong S_n$ for some $n$. This was quite surprising to me!

Noah Schweber
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