Subgroups of free Burnside groups of odd nonprime exponent (Q923130)

Let B be a periodic group, \(B\geq G\neq \{1\}\). The author defines the periodic radical \(\sqrt{G}\) of the subgroup G in B as the subgroup generated by all elements \(b\in B\) such that \(1\neq b^ k\in G\) for some natural k. If \(\sqrt{G}=G\), then the author calls G periodically isolated in B. Now, let \(B=B(\Omega,n)\) be a free Burnside group of odd nonprime exponent \(n>2\cdot 10^{77}\) with a free base of cardinality \(\Omega\) and \(B\geq G\neq \{1\}\). It is proved that if \(n\neq p^ k\) for some prime p, then \(\sqrt{G}=N^*(G)\), where \(N^*(G)=gr\{N(H)|\) \(1\neq H\leq G\}\), and if \(n=p^ k\), then \(\sqrt{G}\leq N^*(G)\leq \sqrt{\sqrt{G}}\). In any case \(\sqrt{G}=G\) iff \(N^*(G)=G\); and this condition always holds if G is a free subgroup of B. In particular, a proper subgroup \(G\leq B\) can not be free if \(G\neq N(G)\) or if G contains some nontrivial normal subgroup of B.