Cardinal invariants distinguishing permutation groups

TL;DR

The paper demonstrates that for infinite cardinals, the alternating group cannot be embedded into the symmetric group of smaller cardinality, using new monotone cardinal invariants.

Contribution

Introduces and studies novel monotone cardinal group invariants to distinguish between permutation groups of different infinite cardinalities.

Findings

$Alt(\lambda)$ is not embeddable into $Sym(\kappa)$ for $\kappa<\\lambda$

Develops new invariants that differentiate permutation groups by cardinality

Provides a method to distinguish permutation groups using these invariants

Abstract

We prove that for infinite cardinals $\kappa<\lambda$ the alternating group $Alt(\lambda)$ (of even permutations) of $\lambda$ is not embeddable into the symmetric group $Sym(\kappa)$ (of all permutations) of $\kappa$. To prove this fact we introduce and study several monotone cardinal group invariants which take value $\kappa$ on the groups $Alt(\kappa)$ and $Sym(\kappa)$.