Locally soluble groups with the restrictions on the generalized norms

Лукашова, Тетяна

doi:10.12958/adm1527

Cited by 3 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

Друшляк

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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Infinite locally finite groups groups with the given properties of the norm of Abelian non-cyclic subgroups

Lukashova,

Drushlyak

2024

Mat. Stud.

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In the paper, the properties of infinite locally finite groups with non-Dedekind locally nil\-potent norms of Abelian non-cyclic subgroups are studied. It is proved that such groups are finite extensions of a quasicyclic subgroup and contain Abelian non-cyclic $p$-subgroups for a unique prime $p$. In particular, in the paper is prove the following assertions: 1) Let $G$ be an infinite locally finite group and contain the locally nilpotent norm $N_{G}^{A}$ with the non-Hamiltonian Sylow $p$-subgroup $(N_{G}^{A})_{p}$. Then $G$ is a finite extension of a quasicyclic $p$-subgroup, all Sylow $p'$-subgroups are finite and do not contain Abelian non-cyclic subgroups. In particular, Sylow $q$-subgroups ($q$ is an odd prime, $q\in \pi(G)$, $q\neq p$) are cyclic, Sylow $2$-subgroups ($p\neq 2$) are either cyclic or finite quaternion $2$-groups (Theorem 1). 2) Let $G$ be a locally finite non-locally nilpotent group with the infinite locally nilpotent non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups. Then $G=G_{p} \leftthreetimes H,$ where $G_{p}$ is an infinite $\overline{HA}_{p}$-group of one of the types (1)--(4) of Proposition~2 in present paper, which coincides with the Sylow $p$-subgroup of the norm $N_{G}^{A}$, $H$ is a finite group, all Abelian subgroups of which are cyclic, and $(|H|,p)=1$. Any element $h\in H$ of odd order that centralizes some Abelian non-cyclic subgroup $M\subset N_{G}^{A}$ is contained in the centralizer of the norm $N_{G}^{A}$. (Theorem 2).3) Let $G$ be an infinite locally finite non-locally nilpotent group with the finite nilpotent non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups. Then$G=H\leftthreetimes K,$ where $H$ is a finite group, all Abelian subgroups of which are cyclic,$\left(\left|H\right|,2\right)=1$, $K$ is an infinite 2-group of one of the types (5)--(6) of Proposition~2 (in present paper). Moreover, the norm $N_{K}^{A}$ of Abelian non-cyclic subgroups of the group $K$ is finite, $K\cap N_{G}^{A}=N_{K}^{A}$ and coincides with the Sylow 2-subgroup $(N_{G}^{A})_2$ of the norm $N_{G}^{A}$ of a group $G$.Moreover, any element $h\in H$ of the centralizer of some Abelian non-cyclic subgroup $M \subset N_{G}^{A}$ is contained in the centralizer of the norm $N_{G}^{A}$. (Theorem 4).

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Locally soluble groups with the restrictions on the generalized norms

Cited by 3 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

Infinite locally finite groups groups with the given properties of the norm of Abelian non-cyclic subgroups

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