On normal subgroups in the periodic products of S.I. Adian

A subgroup H of a given group G is called a hereditarily factorizable subgroup (HF subgroup) if each congruence on H can be extended to some congruence on the entire group G. An arbitrary group G1 is an HF subgroup of the direct product G1 × G2, as well as of the free product G1 * G2 of groups G1 and G2. In this paper a necessary and sufficient condition is obtained for a factor Gi of Adian’s n-periodic product Πi∈InGi of an arbitrary family of groups {Gi}i∈I to be an HF subgroup. We also prove that for each odd n ≥ 1003 any noncyclic subgroup of the free Burnside group B(m, n) contains an HF subgroup isomorphic to the group B(∞, n) of infinite rank. This strengthens the recent results of A.Yu. Ol’shanskii and M. Sapir, D. Sonkin, and S. Ivanov on HF subgroups of free Burnside groups. This result implies, in particular, that each (noncyclic) subgroup of the group B(m, n) is SQ-universal in the class of all groups of period n. Moreover, it turns out that any countable group of period n is embedded in some 2-generated group of period n, which strengthens the previously obtained result of V. Obraztsov. At the end of the paper we prove that the group B(m, n) is distinguished as a direct factor in any n-periodic group in which it is contained as a normal subgroup.