Romanovskij, Nicolai S. On pro-\(p\)-groups with a single defining relator. (English) Zbl 0790.20046 Isr. J. Math. 78, No. 1, 65-73 (1992). 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 \(\text{cd }G \leq 2\). Reviewer: Yu.N.Mukhin (Sverdlovsk) Cited in 3 Documents MSC: 20E18 Limits, profinite groups 20F05 Generators, relations, and presentations of groups 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 20F14 Derived series, central series, and generalizations for groups Keywords:cohomological dimension; free pro-\(p\)-group; defining relator; derived series; torsion free; group algebra PDFBibTeX XMLCite \textit{N. S. Romanovskij}, Isr. J. Math. 78, No. 1, 65--73 (1992; Zbl 0790.20046) Full Text: DOI References: [1] Gildenhuys, D., On pro-p-groups with a single defining relator, Invent. Math., 5, 357-366 (1968) · Zbl 0159.30601 · doi:10.1007/BF01389782 [2] Romanovskii, N., A generalized Freiheitssatz for pro-p-groups, Sib. Math. Zh., 27, 154-170 (1986) · Zbl 0599.20038 [3] Labute, J., Algèbres de Lie et pro-p-groupes définis par une seule relation, Invent. Math., 4, 142-158 (1967) · Zbl 0212.36303 · doi:10.1007/BF01425247 [4] Lazard, M., Sur les groupes nilpotents et les anneaux de Lie, Annales Ecole Normale Supérieure, 71, 101-190 (1954) · Zbl 0055.25103 [5] Remeslennikov, V., Embedding theorems for profinite groups, Izv. Akad. Nauk SSSR, Ser. Mat., 43, 399-417 (1979) · Zbl 0414.20027 [6] Romanovskii, N., On some algorithmic problems for soluble groups, Algebra i Logika, 13, 26-34 (1974) · Zbl 0292.20026 [7] Brumer, A., Pseudocompact algebras, profinite groups and class formations, J. Algebra, 4, 442-470 (1966) · Zbl 0146.04702 · doi:10.1016/0021-8693(66)90034-2 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.