The Lawvere Theory and Monad for the category of Symmetric groups

Question

Take the category of Symmetric groups. Each object is the group of all permutations for some set. Is there a Lawvere theory for this category? What is the monad for the Lawvere theory?

Answer 1

There is no Lawvere theory whose category of models is the category of symmetric groups. This is immediate, for instance, from the fact that the category of models of any Lawvere theory is complete and cocomplete, but the category of symmetric groups has very few limits or colimits (for instance, it does not have a product $S_2\times S_3$ since such a product would have exactly $2\cdot 4=8$ homomorphisms from $S_2$ but there is no symmetric group with that property).