Locally soluble groups with the restrictions on the generalized norms

Лукашова, Тетяна

doi:10.12958/adm1527

Cited by 2 publications

(4 citation statements)

References 11 publications

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“…Due to the rather close relations between the classes of decomposable Abelian and Abelian non-cyclic subgroups the relations between the norms of decomposable and Abelian non-cyclic subgroups were studied in [72,73,89,92]. In [72], such relations were considered for the class of locally őnite groups.…”

Section: Theorem 10 In a Non-periodic Groupmentioning

confidence: 99%

“…Moreover, N d G = (A × ⟨b⟩) ⋋ ⟨c⟩. The study of the impact of the norm N d G of decomposable subgroups on the properties of torsion locally soluble groups was continued in [73,81,92].…”

Section: Theorem 15 ([81]) In a Torsion Locally Nilpotent Group G The...mentioning

confidence: 99%

“…The study of relations between the norm N d G of decomposable subgroups and the norm N A G of Abelian non-cyclic subgroups in torsion groups were continued in [92]. In particular, the properties of torsion groups with identity intersection of these norms were considered.…”

Section: Theorem 15 ([81]) In a Torsion Locally Nilpotent Group G The...mentioning

confidence: 99%

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Generalized norms of groups: retrospective review and current status

Лукашова

Drushlyak

2022

ADM

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Section: Theorem 10 In a Non-periodic Groupmentioning

confidence: 99%

Section: Theorem 15 ([81]) In a Torsion Locally Nilpotent Group G The...mentioning

confidence: 99%

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Generalized norms of groups: retrospective review and current status

Лукашова

Drushlyak

2022

ADM

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“…Narrowing the system of subgroups one can get different Σ-norms which can be considered as generalizations of the norm N (G). Recently the interest to study the Σ-norms does not decrease in view of series of papers [2,3,4,7,8,9,10,11,12,13].…”

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confidence: 99%

Non-periodic groups with the restrictions on the norm of cyclic subgroups of non-prime order

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Лукашова

2022

Mat. Stud.

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One of the main directions in group theory is the study of the impact of characteristic subgroups on the structure of the whole group. Such characteristic subgroups include different $\Sigma$-norms of a group. A $\Sigma$-norm is the intersection of the normalizers of all subgroups of a system $\Sigma$. The authors study non-periodic groups with the restrictions on such a $\Sigma$-norm -- the norm $N_{G}(C_{\bar{p}})$ of cyclic subgroups of non-prime order, which is the intersection of the normalizers of all cyclic subgroups of composite or infinite order of $G$. It was proved that if $G$ is a mixed non-periodic group, then its norm $N_{G}(C_{\bar{p}})$ of cyclic subgroups of non-prime order is either Abelian (torsion or non-periodic) or non-periodic non-Abelian. Moreover, a non-periodic group $G$ has the non-Abelian norm $N_{G}(C_{\bar{p}})$of cyclic subgroups of non-prime order if and only if $G$ is non-Abelian and every cyclic subgroup of non-prime order of a group $G$ is normal in it, and $G=N_{G}(C_{\bar{p}})$.Additionally the relations between the norm $N_{G}(C_{\bar{p}})$ of cyclic subgroups of non-prime order and the norm $N_{G}(C_{\infty})$ of infinite cyclic subgroups, which is the intersection of the normalizers of all infinite cyclic subgroups, in non-periodic groups are studied. It was found that in a non-periodic group $G$ with the non-Abelian norm $N_{G}(C_{\infty})$ of infinite cyclic subgroups norms $N_{G}(C _{\infty})$ and $N_{G}(C _{\bar{p}})$ coincide if and only if $N_{G}(C _{\infty})$ contains all elements of composite order of a group $G$ and does not contain non-normal cyclic subgroups of order 4.In this case $N_{G}(C_{\bar {p}})=N_{G}(C_{\infty})=G$.

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Locally soluble groups with the restrictions on the generalized norms

Cited by 2 publications

References 11 publications

Generalized norms of groups: retrospective review and current status

Generalized norms of groups: retrospective review and current status

Non-periodic groups with the restrictions on the norm of cyclic subgroups of non-prime order

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