具有非素数阶循环子群范数约束的非周期群

Q3 Mathematics
M. Drushlyak, T. Lukashova
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引用次数: 0

摘要

群论的主要方向之一是研究特征子群对整个群结构的影响。这样的特征子群包括一个群的不同$\Sigma$范数。$\Sigma$范数是系统$\Sigma$的所有子群的归一化器的交集。本文研究了非素阶循环子群的范数$N_{G}(C_{\bar{p}})$,它是$G$的复合或无限阶循环子群正规化子的交集。证明了如果$G$是混合非周期群,则其非素数阶循环子群的范数$N_。此外,非周期群$G$具有非素数阶循环子群的非阿贝尔范数$N_,以及$G=N_{G}(C_{\bar{p}})$。此外,还研究了非素数阶循环子群的范数$N_{G}。发现在具有无限循环子群的非阿贝尔范数$N_[infty})=G$。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Non-periodic groups with the restrictions on the norm of cyclic subgroups of non-prime order
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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来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
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