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

V. S. Atabekyan

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algorithmic aspects of algebra and logic, Tome 274 (2011), pp. 15-31

Résumé

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 $G_1$ is an HF subgroup of the direct product $G_1\times G_2$, as well as of the free product $G_1*G_2$ of groups $G_1$ and $G_2$. In this paper a necessary and sufficient condition is obtained for a factor $G_i$ of Adian's $n$-periodic product $\prod_{i\in I}^nG_i$ of an arbitrary family of groups $\{G_i\}_{i\in I}$ to be an HF subgroup. We also prove that for each odd $n\geq1003$ any noncyclic subgroup of the free Burnside group $B(m,n)$ contains an HF subgroup isomorphic to the group $B(\infty,n)$ 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.