On the Root-class Residuality of HNN-extensions of Groups
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...
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Yaroslavl State University
2014-08-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/105 |
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| author | E. A. Tumanova |
| author_facet | E. A. Tumanova |
| author_sort | E. A. Tumanova |
| collection | DOAJ |
| description | 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). |
| format | Article |
| id | doaj-art-1ededc0683ec477e9fcbe5fe6b6a413a |
| institution | Kabale University |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2014-08-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-1ededc0683ec477e9fcbe5fe6b6a413a2025-08-20T03:44:18ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172014-08-0121414818010.18255/1818-1015-2014-4-148-18099On the Root-class Residuality of HNN-extensions of GroupsE. A. Tumanova0Ivanovo State UniversityLet 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).https://www.mais-journal.ru/jour/article/view/105hnn-extensionroot class of groupsroot-class residuality |
| spellingShingle | E. A. Tumanova On the Root-class Residuality of HNN-extensions of Groups Моделирование и анализ информационных систем hnn-extension root class of groups root-class residuality |
| title | On the Root-class Residuality of HNN-extensions of Groups |
| title_full | On the Root-class Residuality of HNN-extensions of Groups |
| title_fullStr | On the Root-class Residuality of HNN-extensions of Groups |
| title_full_unstemmed | On the Root-class Residuality of HNN-extensions of Groups |
| title_short | On the Root-class Residuality of HNN-extensions of Groups |
| title_sort | on the root class residuality of hnn extensions of groups |
| topic | hnn-extension root class of groups root-class residuality |
| url | https://www.mais-journal.ru/jour/article/view/105 |
| work_keys_str_mv | AT eatumanova ontherootclassresidualityofhnnextensionsofgroups |