I am reading Calculus on Manifolds, and just out of curiosity, for $T \in J^{k}(V)$, $S \in J^{l}(V)$ does there exist any relationships between $\operatorname{Alt}(T \otimes S)$ and $\operatorname{Alt(T)} \otimes \operatorname{Alt}(S)$?
I am not really familiar with abstract algebra or permutation groups, so I'm not sure how permutation groups with degree k+l are related to two permutation groups each with degree k and l. Nonetheless, I am still curious if there are any relationships between these two things as it might be useful sometime.