On the nilpotent multiplier of a free product. (Q1413667)

Let \(\mathcal V\) be the variety of groups defined by the set of laws \(V\) and let \(G\) be a group with the free presentation \(1\to R\to F\to G\), then the `Baer invariant' of \(G\) with respect to \(\mathcal V\) is defined to be \({\mathcal V}M(G)=(R\cap V(F))/[RV^*F]\) where \([RV^*F]\) is generated by all elements of the form \(v(f_1,\dots,f_ir,\dots,f_n)v(f_1,\dots,f_i,\dots,f_n)^{-1}\) where \(v\in V\), \(r\in R\) and \(f_j\in F\). \({\mathcal V}M(-)\) is a covariant functor from the category of all groups to the category of Abelian groups, and the main theorem of this paper shows that, when \(\mathcal V\) is the variety of groups of nilpotency class at most \(c\), then this functor commutes with the co-product of cyclic groups of mutually prime order.