Cardinal invariants distinguishing permutation groups

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 $Sym(\kappa )$.