Wiegold, James A non-commutative free algebra of rank 0. (English) Zbl 0625.08006 Rend. Semin. Mat. Univ. Padova 77, 207-211 (1987). Divisible groups may be regarded as the elements of a certain variety \({\mathfrak D}\) of (universal) algebras. An algebra A belongs to \({\mathfrak D}\) if it is a group with additional unary operations \(\rho_ 2,\rho_ 3,..\). satisfying the laws \((x\rho_ n)^ n=x\) for all \(x\in A\). The author investigates the group-theoretic structure of the free algebra F of rank 0 in the variey \({\mathfrak D}\). He shows that the derived series of F is infinite and that F contains non-abelian free subgroups. He also shows that F/F’ is periodic, and, for each prime p, the Sylow p-subgroup of F/F’ is a divisible p-group of countably infinite rank. He conjectures that F is residually soluble. Reviewer: R.M.Bryant MSC: 08B20 Free algebras 20E10 Quasivarieties and varieties of groups 20E07 Subgroup theorems; subgroup growth 20F50 Periodic groups; locally finite groups Keywords:periodic groups; Divisible groups; unary operations; free algebra; derived series; free subgroups; Sylow p-subgroup; divisible p-group PDFBibTeX XMLCite \textit{J. Wiegold}, Rend. Semin. Mat. Univ. Padova 77, 207--211 (1987; Zbl 0625.08006) Full Text: Numdam EuDML References: [1] G.D. Birkhoff , On the structure of abstract algebras , Proc. Cambridge Philos Soc. , 31 ( 1935 ), pp. 433 - 454 . JFM 61.1026.07 · JFM 61.1026.07 [2] M.I. Kargapolov - Yu I. Merzlyakov - V.N. Remeslennikov , Completion of groups , Dokl. Akad. Nauk SSSR , 134 ( 1960 ), pp. 518 - 520 . MR 125160 | Zbl 0097.01402 · Zbl 0097.01402 [3] J.D. Ledlie , Representations of free metabelian D\pi -groups , Trans. Amer. Math. Soc. , 153 ( 1971 ), pp. 307 - 446 . Zbl 0216.08303 · Zbl 0216.08303 · doi:10.2307/1995561 [4] B.H. Neumann - E.C. Wiegold , A semigroup representation of varieties of algebras , Coll. Math. , 14 ( 1966 ), pp. 111 - 114 . MR 191984 | Zbl 0192.09602 · Zbl 0192.09602 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.