Residual Properties of Nilpotent Groups
Let π be a set of primes. Recall that a group G is said to be a residually finite π-group if for every nonidentity element a of G there exists a homomorphism of the group G onto some finite π-group such that the image of the element a differs from 1. A group G will be said to be a virtually residual...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2015-04-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/237 |
| _version_ | 1797877858913222656 |
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| author | D. N. Azarov |
| author_facet | D. N. Azarov |
| author_sort | D. N. Azarov |
| collection | DOAJ |
| description | Let π be a set of primes. Recall that a group G is said to be a residually finite π-group if for every nonidentity element a of G there exists a homomorphism of the group G onto some finite π-group such that the image of the element a differs from 1. A group G will be said to be a virtually residually finite π-group if it contains a finite index subgroup which is a residually finite π-group. Recall that an element g in G is said to be π-radicable if g is an m-th power of an element of G for every positive π-number m. Let N be a nilpotent group and let all power subgroups in N are finitely separable. It is proved that N is a residually finite π-group if and only if N has no nonidentity π-radicable elements. Suppose now that π does not coincide with the set Π of all primes. Let π 0 be the complement of π in the set Π. And let T be a π 0 component of N i.e. T be a set of all elements of N whose orders are finite π 0 -numbers. We prove that the following three statements are equivalent: (1) the group N is a virtually residually finite π-group; (2) the subgroup T is finite and quotient group N/T is a residually finite π-group; (3) the subgroup T is finite and T coincides with the set of all π-radicable elements of N. |
| first_indexed | 2024-04-10T02:24:37Z |
| format | Article |
| id | doaj.art-26389e1616174dfbbf4ba273d4b89240 |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2024-04-10T02:24:37Z |
| publishDate | 2015-04-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-26389e1616174dfbbf4ba273d4b892402023-03-13T08:07:33ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172015-04-0122214915710.18255/1818-1015-2015-2-149-157230Residual Properties of Nilpotent GroupsD. N. Azarov0Ивановский государственный университетLet π be a set of primes. Recall that a group G is said to be a residually finite π-group if for every nonidentity element a of G there exists a homomorphism of the group G onto some finite π-group such that the image of the element a differs from 1. A group G will be said to be a virtually residually finite π-group if it contains a finite index subgroup which is a residually finite π-group. Recall that an element g in G is said to be π-radicable if g is an m-th power of an element of G for every positive π-number m. Let N be a nilpotent group and let all power subgroups in N are finitely separable. It is proved that N is a residually finite π-group if and only if N has no nonidentity π-radicable elements. Suppose now that π does not coincide with the set Π of all primes. Let π 0 be the complement of π in the set Π. And let T be a π 0 component of N i.e. T be a set of all elements of N whose orders are finite π 0 -numbers. We prove that the following three statements are equivalent: (1) the group N is a virtually residually finite π-group; (2) the subgroup T is finite and quotient group N/T is a residually finite π-group; (3) the subgroup T is finite and T coincides with the set of all π-radicable elements of N.https://www.mais-journal.ru/jour/article/view/237нильпотентная группагруппа конечного рангааппроксимируемость конечными p-группами |
| spellingShingle | D. N. Azarov Residual Properties of Nilpotent Groups Моделирование и анализ информационных систем нильпотентная группа группа конечного ранга аппроксимируемость конечными p-группами |
| title | Residual Properties of Nilpotent Groups |
| title_full | Residual Properties of Nilpotent Groups |
| title_fullStr | Residual Properties of Nilpotent Groups |
| title_full_unstemmed | Residual Properties of Nilpotent Groups |
| title_short | Residual Properties of Nilpotent Groups |
| title_sort | residual properties of nilpotent groups |
| topic | нильпотентная группа группа конечного ранга аппроксимируемость конечными p-группами |
| url | https://www.mais-journal.ru/jour/article/view/237 |
| work_keys_str_mv | AT dnazarov residualpropertiesofnilpotentgroups |