On the Neumann-Shmel'kin theorem
The theorem of the title asserts that any nontrivial proper variety of groups has a unique decomposition as a finite product of indecomposable varieties. The present paper considers the relationship between the product decompositions of a variety of groups (not necessarily into indecomposables) and those of a subvariety. Theorem: Let \(U\subseteq V\) be varieties of groups with decompositions \(U=U_ 1...U_ k\), \(V=V_ 1...V_{\ell}\). Then there exists an intermediate variety M, \(U\subseteq M\subseteq V\), with two decompositions \(M=M_ 1...M_ k=N_ 1...N_{\ell}\) such that \(U_ i\subseteq M_ i\) for each i and \(N_ j\subseteq V_ j\) for each j.
scientific article; zbMATH DE number 4002368
scientific article