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A non-commutative free algebra of rank 0. (English) Zbl 0625.08006

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
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References:

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