On the structure of groups, whose subgroups are either normal or core-free
We investigate the influence of some natural types of subgroups on the structure of groups. A subgroup H of the group G is called core-free if CoreG(H) = 〈1〉. We study the groups, in which every subgroup is either normal or core-free. More precisely, we obtain the structures of monolithic and non-...
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Kurdachenko, L.A. Pypka, A.A. Subbotin, I.Ya. 2019-07-12T19:33:37Z 2019-07-12T19:33:37Z 2019 On the structure of groups, whose subgroups are either normal or core-free / L.A. Kurdachenko, A.A. Pypka, I.Ya. Subbotin // Доповіді Національної академії наук України. — 2019. — № 4. — С. 17-20. — Бібліогр.: 8 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2019.04.017 https://nasplib.isofts.kiev.ua/handle/123456789/158089 512.544 We investigate the influence of some natural types of subgroups on the structure of groups. A subgroup H of the group G is called core-free if CoreG(H) = 〈1〉. We study the groups, in which every subgroup is either normal or core-free. More precisely, we obtain the structures of monolithic and non-monolithic groups with this property. Досліджується вплив деяких природних типів підгруп на структуру груп. Підгрупу H групи G називаємо вільною від ядра, якщо CoreG(H) = 〈1〉 . Вивчено групи, в яких кожна підгрупа або нормальна, або вільна від ядра. Точніше, одержано будову монолітичних та немонолітичних груп з цією властивістю. Исследуется влияние некоторых естественных типов подгрупп на структуру групп. Подгруппу H группы G называем свободной от ядра, если CoreG(H) = 〈1〉 . Изучены группы, в которых каждая подгруппа либо нормальна, либо свободна от ядра. Точнее, получена структура монолитических и немонолитических групп с этим свойством. en Видавничий дім "Академперіодика" НАН України Доповіді НАН України Математика On the structure of groups, whose subgroups are either normal or core-free Про структуру груп, підгрупи яких або нормальні, або вільні від ядра О структуре групп, подгруппы которых либо нормальны, либо свободны от ядра Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On the structure of groups, whose subgroups are either normal or core-free |
| spellingShingle |
On the structure of groups, whose subgroups are either normal or core-free Kurdachenko, L.A. Pypka, A.A. Subbotin, I.Ya. Математика |
| title_short |
On the structure of groups, whose subgroups are either normal or core-free |
| title_full |
On the structure of groups, whose subgroups are either normal or core-free |
| title_fullStr |
On the structure of groups, whose subgroups are either normal or core-free |
| title_full_unstemmed |
On the structure of groups, whose subgroups are either normal or core-free |
| title_sort |
on the structure of groups, whose subgroups are either normal or core-free |
| author |
Kurdachenko, L.A. Pypka, A.A. Subbotin, I.Ya. |
| author_facet |
Kurdachenko, L.A. Pypka, A.A. Subbotin, I.Ya. |
| topic |
Математика |
| topic_facet |
Математика |
| publishDate |
2019 |
| language |
English |
| container_title |
Доповіді НАН України |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| format |
Article |
| title_alt |
Про структуру груп, підгрупи яких або нормальні, або вільні від ядра О структуре групп, подгруппы которых либо нормальны, либо свободны от ядра |
| description |
We investigate the influence of some natural types of subgroups on the structure of groups. A subgroup H of the group
G is called core-free if CoreG(H) = 〈1〉. We study the groups, in which every subgroup is either normal or core-free.
More precisely, we obtain the structures of monolithic and non-monolithic groups with this property.
Досліджується вплив деяких природних типів підгруп на структуру груп. Підгрупу H групи G називаємо
вільною від ядра, якщо CoreG(H) = 〈1〉 . Вивчено групи, в яких кожна підгрупа або нормальна, або вільна
від ядра. Точніше, одержано будову монолітичних та немонолітичних груп з цією властивістю.
Исследуется влияние некоторых естественных типов подгрупп на структуру групп. Подгруппу H группы
G называем свободной от ядра, если CoreG(H) = 〈1〉 . Изучены группы, в которых каждая подгруппа либо
нормальна, либо свободна от ядра. Точнее, получена структура монолитических и немонолитических
групп с этим свойством.
|
| issn |
1025-6415 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/158089 |
| citation_txt |
On the structure of groups, whose subgroups are either normal or core-free / L.A. Kurdachenko, A.A. Pypka, I.Ya. Subbotin // Доповіді Національної академії наук України. — 2019. — № 4. — С. 17-20. — Бібліогр.: 8 назв. — англ. |
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2025-11-24T02:44:21Z |
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| fulltext |
17ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2019. № 4
doi: https://doi.org/10.15407/dopovidi2019.04.017
UDC 512.544
L.A. Kurdachenko 1, A.A. Pypka 1, I.Ya. Subbotin 2
1 Oles Honchar Dnipro National University, Ukraine
2 National University, Los Angeles, USA
E-mail: lkurdachenko@i.ua, pypka@ua.fm, isubboti@nu.edu
On the structure of groups, whose subgroups
are either normal or core-free
Presented by Corresponding Member of the NAS of Ukraine V.P. Motornyi
We investigate the influence of some natural types of subgroups on the structure of groups. A subgroup H of the group
G is called core-free if Core ( )G H = 〈1〉. We study the groups, in which every subgroup is either normal or core-free.
More precisely, we obtain the structures of monolithic and non-monolithic groups with this property.
Keywords: normal subgroup, core-free subgroup, Dedekind group.
Let G be a group. The following two normal subgroups are associated with any subgroup H of the
group G: HG, the normal closure of H in a the group G, the least normal subgroup of G including
H, and Core ( )G H , the (normal) core of H in G, the greatest normal subgroup of G, which is
contained in H. We have
|G xH H x G= 〈 ∈ 〉
and
Core ( ) x
G
x G
H H
∈
= ∩ .
A subgroup H is normal if and only if Core ( )G
GH H H= = . In this sense, the subgroups H, for
which Core ( )G H = 〈1〉 , are the complete opposite of the normal subgroups. A subgroup H of the
group G is called core-free in G if Core ( )G H = 〈1〉 .
There is a whole series of papers devoted to the study of groups with only two types of sub-
groups: subgroups with some property ρ and subgroups with a property that is antagonistic to
ρ (see, for example, [1—6]). In particular, from the results of paper [3], it is possible to obtain a
description of groups that have only two possibilities for each subgroup H: GH H= or GH G= .
In this connection, a dual question naturally arises on the structure of groups, in which, for each
subgroup H, there are only two other possibilities: Core ( )G H H= or Core ( )G H = 〈1〉 . The finite
groups having this property had been studied in [7]. Note at once that the groups, whose all sub-
groups are normal, possess this property.
© L.A. Kurdachenko, A.A. Pypka, I.Ya. Subbotin, 2019
18 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2019. № 4
L.A. Kurdachenko, A.A. Pypka, I.Ya. Subbotin
Recall that a group G is called Dedekind, if every its subgroup is normal. The Dedekind group
G has the following structure: it is either Abelian or 8G Q D P= × × , where 8Q is a quaternion
group of order 8, D is an elementary Abelian 2-group, and P is an Abelian 2′-group [8].
Another extreme case that occurs here is the simple groups. In them, every proper subgroup
is core-free. This fact immediately shows that the study of groups, in which Core ( )G H H= or
Core ( )G H = 〈1〉 for each subgroup H, makes sense for generalized soluble groups. The two key
cases here are as follows: G is a non-monolithic group or G is a monolithic group. Let G be a group.
The intersection of all non-trivial normal subgroups Mon( )G of G is called the monolith of the
group G. If Mon( )G ≠ 〈1〉 , then the group G is called monolithic, and, in this case, Mon( )G is the
least non-trivial normal subgroup of G.
Our first main result is related to the non-monolithic case.
Theorem A. Let G be an infinite group, whose non-normal subgroups are core-free. If G is non-
monolithic, then G is a Dedekind group.
The following our main theorem considers the monolithic case. Here, we get a much more
diverse situation. Separate considerations are required for non-periodic and periodic groups.
Theorem B. Let G be a locally soluble non-periodic group, whose non-normal subgroups are core-
free. Suppose that G is not a Dedekind group. Then G is monolithic, the factor-group /Mon( )G G is
non-periodic, Mon( )G G A= , and the following conditions hold:
(i) Mon( )G is either torsion-free Abelian subgroup or elementary Abelian p-subgroup for some
prime p;
(ii) [ , ] Mon( ) (Mon( ))GG G G C G= = ;
(iii) a subgroup A is Abelian, and Tor( )A is locally cyclic;
(iv) if Mon( )G is an elementary Abelian p-subgroup, then Tor( )A is a p′-subgroup;
(v) if A has finite 0-rank, then Mon( )G is an elementary Abelian p-subgroup;
(vi) if B is another complement to Mon( )G in G, then the subgroups A and B are conjugate.
In turn, the case where G is periodic also splits into two cases depending on whether the
center includes a monolith or not. Recall that a p-group G is called extraspecial, if [ , ] ( )G G G= ζ is
a subgroup of order p and / ( )G Gζ is an elementary Abelian p-group.
From this definition, we can see that the center of an extraspecial p-group G is the least nor-
mal subgroup, so that if H is a subgroup of G, and H includes a non-trivial G-invariant subgroup,
then H includes ( )Gζ . The equality [ , ] ( )G G G= ζ implies that H is normal in G. In other words,
every subgroup of G is either normal or core-free.
Theorem C. Let G be a periodic monolithic group, whose non-normal subgroups are core-free.
Suppose that G is not a Dedekind group. If the center of G includes a monolith, then G KE= , where K
is a cyclic or quasicyclic p-subgroup, E is an extraspecial p-subgroup, ( )K G= ζ , and [ , ]K E G G∩ =
is a subgroup of order p, p is a prime.
Theorem D. Let G be an infinite periodic locally soluble monolithic group, whose non-normal
subgroups are core-free. Suppose that G is not a Dedekind group and the monolith of G is not central.
Then Mon( )G G A= , and the following conditions hold:
(i) Mon( )G is an infinite elementary Abelian p-subgroup for some prime p, and A is an infinite
periodic p′-group;
(ii) [ , ] Mon( ) (Mon( ))GG G G C G= = ;
19ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2019. № 4
On the structure of groups, whose subgroups are either normal or core-free
(iii) whether the subgroup A is locally cyclic, or A Q B= × , where Q is a quaternion group of
order 8, and B is a locally cyclic 2′-subgroup;
(iv) if C is another complement to Mon( )G in G, then the subgroups A and C are conjugate.
Note that if /Mon( )G G is finite or Mon( )G is finite and non-central, then G is finite (this
follows from Theorem D). The last our result gives a description of the finite soluble group, whose
non-normal subgroups are core-free. As was noted above, a finite group, whose non-normal sub-
groups are core-free, was studied in [7]. Our description is more detailed than the description
given in Theorem 1 of that paper. We also note that the proof of Lemma 5 in [7] contains a gap
(only the case where the both factor-groups 1/G N and 2/G N are non-Abelian was considered).
In addition, there is a mistake there: the fact that H is a subgroup of T A× does not implies that
1 2H H H= × , where 1H T and 2H A . Therefore, we do not use the results of work [7]. We
proved of the following result.
Theorem E. Let G be a finite soluble group, whose non-normal subgroups are core-free. Suppose
that G is not a Dedekind group. Then G is monolithic.
If the center of G includes a monolith, then G KE= where K is a cyclic p-subgroup, E is an ex-
traspecial p-subgroup, ( )K G= ζ , and [ , ]K E G G∩ = is a subgroup of order p, p is a prime.
If the monolith of G is not central, then Mon( )G G A= , and the following conditions hold:
(i) Mon( )G is elementary Abelian p-subgroup for some prime p, and A is a p′-group;
(ii) [ , ] Mon( ) (Mon( ))GG G G C G= = ;
(iii) whether a subgroup A is cyclic or A Q B= × , where Q is a quaternion group of order 8, and
B is a cyclic 2′-subgroup;
(iv) if C is another complement to Mon( )G in G, then the subgroups A and C are conjugate.
REFERENCES
1. Fattahi, A. (1974). Groups with only normal and abnormal subgroups. J. Algebra, 28, No. 1, pp. 15-19.
2. Ebert, G. & Bauman, S. (1975). A note of subnormal and abnormal chains. J. Algebra, 36, No. 2, pp. 287-293.
3. De Falco, M., Kurdachenko, L.A. & Subbotin, I.Ya. (1998). Groups with only abnormal and subnormal
subgroups. Atti Sem. Mat. Fis. Univ. Modena, 46, pp. 435-442.
4. Kurdachenko, L.A. & Smith, H. (2005). Groups with all subgroups either subnormal or self-normalizing. J.
Pure Appl. Algebra, 196, No. 2-3, pp. 271-278.
5. Kurdachenko, L.A., Otal, J., Russo, A. & Vincenzi, G. (2011). Groups whose all subgroups are ascendant or
self-normalizing. Cent. Eur. J. Math., 9, No. 2, pp. 420-432.
6. Kurdachenko, L.A., Pypka, A.A. & Semko, N.N. (2016). The groups whose cyclic subgroups are either ascendant
or almost self-normalizing. Algebra Discrete Math., 21, No. 1, pp. 111-127.
7. Zhao, L., Li, Y. & Gong, L. (2018). Finite groups in which the cores of every non-normal subgroups are trivial.
Publ. Math. Debrecen, 93, No. 3-4, pp. 511-516.
8. Baer, R. (1933). Situation der Untergruppen und Struktur der Gruppe. S.-B. Heidelberg Acad. Math.-Nat.
Klasse, 2, pp. 12-17.
Received 02.01.2019
20 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2019. № 4
L.A. Kurdachenko, A.A. Pypka, I.Ya. Subbotin
Л.А. Курдаченко 1, О.О. Пипка 1, І.Я. Субботін 2
1 Дніпровський національний університет ім. Олеся Гончара
2 Національний університет, Лос-Анджелес, США
E-mail: lkurdachenko@i.ua, pypka@ua.fm, isubboti@nu.edu
ПРО СТРУКТУРУ ГРУП, ПІДГРУПИ ЯКИХ
АБО НОРМАЛЬНІ, АБО ВІЛЬНІ ВІД ЯДРА
Досліджується вплив деяких природних типів підгруп на структуру груп. Підгрупу H групи G називаємо
вільною від ядра, якщо Core ( )G H = 〈1〉 . Вивчено групи, в яких кожна підгрупа або нормальна, або вільна
від ядра. Точніше, одержано будову монолітичних та немонолітичних груп з цією властивістю.
Ключові слова: нормальна підгрупа, вільна від ядра підгрупа, дедекіндова група.
Л.А. Курдаченко 1, А.А. Пыпка 1, И.Я. Субботин 2
1 Днепровский национальный университет им. Олеся Гончара
2 Национальный университет, Лос-Анджелес, США
E-mail: lkurdachenko@i.ua, pypka@ua.fm, isubboti@nu.edu
О СТРУКТУРЕ ГРУПП, ПОДГРУППЫ КОТОРЫХ
ЛИБО НОРМАЛЬНЫ, ЛИБО СВОБОДНЫ ОТ ЯДРА
Исследуется влияние некоторых естественных типов подгрупп на структуру групп. Подгруппу H группы
G называем свободной от ядра, если Core ( )G H = 〈1〉 . Изучены группы, в которых каждая подгруппа либо
нормальна, либо свободна от ядра. Точнее, получена структура монолитических и немонолитических
групп с этим свойством.
Ключевые слова: нормальная подгруппа, свободная от ядра подгруппа, дедекиндова группа.
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