In this paper, we use the theory of simplicial groups to develop the Schur multiplier of a pair of groups $(G,N)$ to the Baer invariant of it, $\mathcal{ V}M(G,N)$, with respect to an arbitrary variety $\mathcal{ V}$. Moreover, we present among other things some behaviors of Baer invariants of a pair of groups with respect to the free product and the direct limit. Finally we prove that the nilpotent multiplier of a pair of groups does commute with the free product of finite groups of mutually coprime orders.
Categories: math.AT math.GR