Cardinal invariants distinguishing permutation groups

We prove that for infinite cardinals \(\kappa <\lambda \) the alternating group \(\mathrm {Alt}(\lambda )\) (of even permutations) of \(\lambda \) is not embeddable into the symmetric group \(\mathrm {Sym}(\kappa )\) (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 \(\mathrm {Alt}(\kappa )\) and \(\mathrm {Sym}(\kappa )\).