On the Supersoluble Residual of a Product of Supersoluble Subgroups
Let P be the set of all primes. A subgroup H of a group G is called P-subnormal in G, if either H = G, or there exists a chain of subgroups H = H_0 \leq H_1 \leq ... \leq H_n = G, with |H_i : H_{i-1}| \in P for all i. A group G = AB with P-subnormal supersoluble subgroups A and B is studied. T...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Aracne
2020-06-01
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| Series: | Advances in Group Theory and Applications |
| Subjects: | |
| Online Access: | http://www.advgrouptheory.com/journal/Volumes/9/MonakhovTrofimuk.pdf |
| _version_ | 1818340516721328128 |
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| author | Victor S. Monakhov Alexander A. Trofimuk |
| author_facet | Victor S. Monakhov Alexander A. Trofimuk |
| author_sort | Victor S. Monakhov |
| collection | DOAJ |
| description | Let P be the set of all primes. A subgroup H of a group G is called P-subnormal in G, if either H = G, or there exists a chain of subgroups
H = H_0 \leq H_1 \leq ... \leq H_n = G,
with |H_i : H_{i-1}| \in P for all i. A group G = AB with P-subnormal supersoluble subgroups A and B is studied. The structure of its supersoluble residual is obtained. In particular, it coincides with the nilpotent residual of the derived subgroup of G. Besides, if the indices of the subgroups A and B are coprime, then the supersoluble residual coincides with the intersection of the metanilpotent residual of G and all normal subgroups of G such that all corresponding quotients are primary or biprimary. From here new signs of supersolubility are derived. |
| first_indexed | 2024-12-13T15:44:09Z |
| format | Article |
| id | doaj.art-6cef47c3880e426c81adeb3c4323ac9d |
| institution | Directory Open Access Journal |
| issn | 2499-1287 2499-1287 |
| language | English |
| last_indexed | 2024-12-13T15:44:09Z |
| publishDate | 2020-06-01 |
| publisher | Aracne |
| record_format | Article |
| series | Advances in Group Theory and Applications |
| spelling | doaj.art-6cef47c3880e426c81adeb3c4323ac9d2022-12-21T23:39:45ZengAracneAdvances in Group Theory and Applications2499-12872499-12872020-06-019517010.32037/agta-2020-003On the Supersoluble Residual of a Product of Supersoluble SubgroupsVictor S. Monakhov0Alexander A. Trofimuk1Francisk Skorina Gomel State UniversityFrancisk Skorina Gomel State UniversityLet P be the set of all primes. A subgroup H of a group G is called P-subnormal in G, if either H = G, or there exists a chain of subgroups H = H_0 \leq H_1 \leq ... \leq H_n = G, with |H_i : H_{i-1}| \in P for all i. A group G = AB with P-subnormal supersoluble subgroups A and B is studied. The structure of its supersoluble residual is obtained. In particular, it coincides with the nilpotent residual of the derived subgroup of G. Besides, if the indices of the subgroups A and B are coprime, then the supersoluble residual coincides with the intersection of the metanilpotent residual of G and all normal subgroups of G such that all corresponding quotients are primary or biprimary. From here new signs of supersolubility are derived.http://www.advgrouptheory.com/journal/Volumes/9/MonakhovTrofimuk.pdfsupersoluble groupsubnormal subgroupseminormal subgroupp-subnormal subgroupderived subgroupsupersoluble residual |
| spellingShingle | Victor S. Monakhov Alexander A. Trofimuk On the Supersoluble Residual of a Product of Supersoluble Subgroups Advances in Group Theory and Applications supersoluble group subnormal subgroup seminormal subgroup p-subnormal subgroup derived subgroup supersoluble residual |
| title | On the Supersoluble Residual of a Product of Supersoluble Subgroups |
| title_full | On the Supersoluble Residual of a Product of Supersoluble Subgroups |
| title_fullStr | On the Supersoluble Residual of a Product of Supersoluble Subgroups |
| title_full_unstemmed | On the Supersoluble Residual of a Product of Supersoluble Subgroups |
| title_short | On the Supersoluble Residual of a Product of Supersoluble Subgroups |
| title_sort | on the supersoluble residual of a product of supersoluble subgroups |
| topic | supersoluble group subnormal subgroup seminormal subgroup p-subnormal subgroup derived subgroup supersoluble residual |
| url | http://www.advgrouptheory.com/journal/Volumes/9/MonakhovTrofimuk.pdf |
| work_keys_str_mv | AT victorsmonakhov onthesupersolubleresidualofaproductofsupersolublesubgroups AT alexanderatrofimuk onthesupersolubleresidualofaproductofsupersolublesubgroups |