
doi: 10.1007/bf02773840
With every group \(G\) the author associates a family of groups \(\Gamma_ n(G)\) each of which is equal to the \(n\)-th term \(\gamma_ n(U)\) where \(U\) is a projective object in the category of epimorphisms \(U\twoheadrightarrow G\) with the kernel in the \(n\)-th centre of \(U\). It is noticed that these groups coincide with R. Baer's invariants and thus can be applied to the study of the integral homology of \(G\), as well as to some other questions in group theory. Canonically, \(\Gamma_ n(G)\twoheadrightarrow \gamma_ n(G)\) is a central extension and also, for \(m\geq n\), we have canonical homomorphisms \(\Gamma_ m(G)\twoheadrightarrow\Gamma_ n(G)\) which endow each \(\Gamma_ n(G)\) with a central filtration and enable to apply Lie ring techniques developed in this case to a certain extent. As the main application the author presents an exact bound for the rank of \(H_ 2(G)\), \(G\) finitely generated nilpotent, in terms of E. Witt's numbers for the dimensions of homogeneous components in free Lie rings. The author remarks however that this estimate is essentially due to N. Blackburn -- L. Evans, 1979. Further, there are some estimates for the order of \(\Gamma_ n(G)\) which is shown to be finite as soon as \(G\) is finite.
Homological methods in group theory, central filtration, Derived series, central series, and generalizations for groups, Associated Lie structures for groups, Baer's invariants, finitely generated nilpotent, central extension, category of epimorphisms, projective object, Lie ring, free Lie rings, integral homology, homogeneous components
Homological methods in group theory, central filtration, Derived series, central series, and generalizations for groups, Associated Lie structures for groups, Baer's invariants, finitely generated nilpotent, central extension, category of epimorphisms, projective object, Lie ring, free Lie rings, integral homology, homogeneous components
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