Characteristic properties and uniform non-amenability of -periodic products of groups

We prove that -periodic products (introduced by the first author in 1976) are uniquely characterized by certain quite specific properties. Using these properties, we prove that if a non-cyclic subgroup of the -periodic product of a given family of groups is not conjugate to any subgroup of the product's components, then contains a subgroup isomorphic to the free Burnside group . This means that contains the free periodic groups of any rank , which lie in ([1], Russian p. 26). Moreover, if is finitely generated, then it is uniformly non-amenable. We also describe all finite subgroups of -periodic products.