Subgroup theorems for the Baer-invariant of groups (Q1270393)

Let \(F_\infty\) be the free group freely generated by an infinite set \(\{x_1,x_2,\dots\}\). Let \(w\) be an outer commutator word in \(F_\infty\) and assume that \(\mathcal V\) be the variety of groups defined by the commutator word \([w,x_1,x_2]\). Let \(G\) be a finite group with \(\mathcal V\)-stem cover \(G^*/L\cong G\). If \(H\) is a subgroup of \(G\) of finite index \(n\) such that \(H\cong B/L\), then it is shown (Theorem 3.6) that the commutator subgroup \([w(B),{\mathcal V}M(G)^n]\) is isomorphic to a subgroup of \({\mathcal V}M(H)\), where \(w(B)\) is the verbal subgroup of \(B\) with respect to the word \(w\), and \({\mathcal V}M(G)\) is the Baer-invariant of \(G\) with respect to the variety of groups \(\mathcal V\). It is also shown that if \({\mathcal N}_c\) is the variety of nilpotent groups of class at most \(c\) and \(c+1\) is any prime number or 4, \(G\) is a finite nilpotent group with Sylow \(p\)-subgroup \(G_p\), then \({\mathcal N}_cM(G)_p\cong{\mathcal N}_cM(G_p)\) (Theorem 4.3). An example, which shows that these results cannot be further generalized, is constructed.