On the homologies of free soluble groups
The author studies the homology groups of the free soluble group \(\phi_s=F/F^{(n)}\), with coefficients in the field \(\mathbb{Z}_p\) of integers \(\text{mod }p\). If \(\phi^{(k)}_s\) is the \(k\)-th derived group of \(\phi_s\) (the \(k\)th commutant in the translator's wording), then \(\phi_s/\phi_s^{(k)}\) \((k \leq s)\) is also free soluble of (solubility) class \(s-k\). The embedding \(\phi^{(k)}_s \to \phi_s\) induces homomorphisms of homology groups \(h_n^{(k)}: H_n (\Phi_s^{(k)}, \mathbb{Z}_p)\to H_n (\Phi_s, \mathbb{Z}_p)\) and if \(H_n^{(k)}=\text{Im }h_n^{(k)}\), then there is a filtration of \(H_n^{(k)} (\phi_s, \mathbb{Z}_n)\): (1) \(0=H^{(s)}_n \subseteq H_n^{(s-1)} \subseteq \cdots \subseteq H_n^{(1)} \subseteq H^{(0)}_n=H_n (\Phi_s, \mathbb{Z}_p)\). The main result (in the translator's wording) is: Theorem. Suppose \(\Phi_s\) is a free soluble group of length \(s\) and \(M_k=\Phi^{(k- 1)}_s/ \Phi^{(k)}_s\) is the \(k\)-th factor of its commutant series. Denote \(E^{(k)}_n=H_n^{(k-1)}/H^{(k)}_n\), where \(H^{(k)}_n\) is the image of a natural homomorphism \(h^{(k)}_n : H_n(\Phi^{(k)}_s, \mathbb{Z}_p) \to H_n(\Phi_s, \mathbb{Z}_p)\) \((E^{(k)}_n\) are factors of the filtration (1)). Then for \(p \neq 2\) 1) there exists an epimorphism \({\mathcal C}^{s-k}_p (x^n) \Lambda^* M_k \otimes \mathbb{Z}_p \to E^{(k)}_n\), 2) the exact sequence \(0\to x{\mathcal C}^{s-k+1}_p(x^n)H_*(\Phi_{k- 1},\mathbb{Z}_p)\to{\mathcal C}^{s-k}_p (x^n) \Lambda^* M_k \otimes_{\Phi_{k-1}} \mathbb{Z}_p \to E^{(k)}_n \to 0\) holds, and 3) the co-kernel of the mapping \(h^{(k)}_n\) is isomorphic to \({\mathcal C}^{s-k}_p (x^n) H_* (\Phi_k, \mathbb{Z}_p)\). Here \[ {\mathcal C}_p(x^n)=\begin{cases} x^n \varphi^{(p)}_{n-1}, & \text{if \(n=1 \pmod p\), \(n > 1\)},\\0, & \text{otherwise,}\end{cases} \] where \(\varphi^{(p)}_n\) are polynomials defined explicitly. The author discusses some corollaries explaining what kind of information can be obtained from the theorem.
scientific article; zbMATH DE number 795650