On pro-\(p\)-groups with a single defining relator
Let \(F\) be a free pro-\(p\)-group with a basis \(X\) and \(G = \langle X\mid r\rangle\) be a pro-\(p\)-group with a single defining relator \(r\in F^{(k)} \setminus F^{(k+1)}\), \(F^{(k)}\) be the \(n\)-th commutator subgroup of \(F\). It is proved that the factors of the derived series of \(G\) are torsion free iff \(r\) is not a \(p\)-th power in \(F^{(k)} / F^{(k+1)}\), in which case the group algebra \(\mathbb{Z}_ pG\) is a domain and \(\text{cd }G \leq 2\).
scientific article; zbMATH DE number 219367
scientific article