Some properties on the Baer-invariant of a pair of groups and \(\mathcal V_G\)-marginal series. (Q2839027)

Let \(1\to R\to F\to G\to 1\) be a free presentation of the group \(G\) and \(N\) be a normal subgroup of \(G\) such that \(N\cong S/R\) for a normal subgroup \(S\) of the free group \(F\). Then, the `Baer-invariant' of the pair of groups \((G,N)\) with respect to a variety \(\mathcal V\), denoted by \(\mathcal VM(G,N)\), is defined by NEWLINE\[NEWLINE\mathcal VM(G,N)=R\cap[SV^*F]/[RV^*F].NEWLINE\]NEWLINE It is known that \(\mathcal VM(G,N)\) is an Abelian group and independent of the choice of the free presentation of \(G\). In the article under review, the authors present some properties of the Baer-invariant of a pair of groups with respect to a given variety of groups \(\mathcal V\). In particular, they derive some equalities and inequalities of the Baer-invariant of a pair of finite groups, as long as \(\mathcal V\) is considered to be a Schur-Baer variety. Moreover, they present a relative version of the concept of lower marginal series and give some isomorphisms among \(\mathcal V_G\)-marginal factor groups. Also, they present a generalized version of Stallings' theorem.