Cardinal invariants distinguishing permutation groups

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

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