On the Root-class Residuality of HNN-extensions of Groups - E. A. Tumanova (Моделирование и анализ информационных систем, 2014)
Let K be an arbitrary root class of groups. This means that K contains at least one non-unit group, is closed under taking subgroups and direct products of a finite number of factors and satisfies the Gruenberg condition: if 1 ≤ Z ≤ Y ≤ X is a subnormal series of a group X such that X/Y ∈ K and Y/Z ∈ K, there exists a normal subgroup T of X such that T ⊆ Z and X/T ∈ K. In this paper we study the property ‘to be residually a K-group’ of an HNN-extension in the case when its associated subgroups coincide. Let G = (B, t; t¯¹Ht = H, φ). We get a sufficient condition for G to be residually a K-group in the case when B ∈ K and H is normal in B, which turns out to be necessary if K is closed under factorization. We also obtain criteria for G to be residually a K-group provided that K is closed under factorization, B is residually a K-group, H is normal in B and satisfies at least one of the following conditions: AutG(H) is abelian (we denote by AutG(H) the group of all automorphisms of H which are the restrictions on this subgroup of all inner automorphisms of G); AutG(H) is finite; φ coincides with the restriction on H of an inner automorphism of B; H is finite; H is infinite cyclic; H is of finite Hirsh-Zaitsev rank (i. e. H possesses a finite subnormal series all factors of which are either periodic or infinite cyclic). Besides, we find a sufficient condition for G to be residually a K-group in the case when B is residually a K-group and H is a retract of B (K is not necessarily closed under the factorization in this statement).