I'm learning a little bit about set theory and weird cardinals and I'm trying to come up with ways to construct familiar objects from abstract algebra out of cardinals. Limiting the number of fixed points in every permutation seems like an obvious thing to do, so I'm wondering if this construction has a name so I can read more about it.
Given any set $X$, we construct the symmetric group on $X$ by considering all permutations of $X$ together with composition as our operation.
Now let's take an infinite cardinal $\kappa$ and consider the groups $\text{Sym}_{< \kappa}(X)$ and $\text{Sym}_{\le \kappa}(X)$, defined as follows:
$$ \text{Sym}_{\lt \kappa}(X) := \\ \{ \text{$z \in \text{Sym}(X)$ and the number of points not fixed by $z$ is strictly less than $\kappa$} \} $$
And likewise for $\text{Sym}_{\le \kappa}(X)$. It's not hard to show that $\text{Sym}_{\le \kappa}(X) = \text{Sym}_{\lt \kappa^+}(X)$ .
I know from this comment from Alex Kruckman on this question I asked a few years ago that I can't distinguish $\kappa$ and $\lambda$ in general by comparing whether $\text{Sym}(\kappa)$ and $\text{Sym}(\lambda)$ are isomorphic without additional background assumptions. Although, admittedly, I don't understand why this is independent of ZFC.
However, I can distinguish $\kappa$ and $\lambda$ by using $\text{Sym}_{< \omega}(\kappa)$ and $\text{Sym}_{< \omega}(\lambda)$, since each of those groups has the same cardinality as their carrier.
Proof.
- The size of $\text{Sym}_{< \omega}(\zeta)$ is bounded below by the set of all transpositions of elements of $\zeta$, which has the same size as $|\zeta \times \zeta| = |\zeta|$.
- The size of $\text{Sym}_{< \omega}(\zeta)$ is bounded above by finite sequences of transpositions in $\zeta$, which has the same size as $\zeta$.
