On periodic Products of Groups

Adian introduced periodic n-products of groups which are given by imposing of defining relations of the form An=1 on the free product of groups Gα, α∈I, without involutions. The defining relations An=1 are constructed by a complicated induction which is quite similar to the inductive construction of free Burnside groups due to Novikov and Adian. This periodic n-product of groups Gα, α∈I, has the remarkable property that for every either xn=1 or x is conjugate to an element of Gα for some α. The main result of the article is that this property of periodic n-product can be taken as its definition. This gives a new non-inductive characterization of periodic n-products. An analogous characterization of periodic -products due to Ol’shanskii is also given.