On products on minimal simplicial sets

We investigate product structures on minimal simplicial sets. If X is a connected minimal simplicial set homotopy equivalent to a free loop space of a finite CW-complex, then we show that there exists a (non-associative) multiplication in X such that X is generated by finitely many elements in the homotopy groups π*(X) X as a group-like simplicial H-set. Some examples of minimal simplicial groups are given.