On weakly $\mathcal{H}$-permutable subgroups of finite groups

Rok, strany: 2019, 763 - 772

Let $σ =\{σ_i |i\in I\}$ be some partition of the set of all primes $\mathbb{P}$, $G$ be a finite group and $σ(G)=\{σ_i|σ_i\cap π(G)\neq \emptyset\}$. $G$ is said to be \emph{$σ$-primary} if $|σ(G)|≤ 1$. A subgroup $H$ of $G$ is said to be \emph{$σ$-subnormal} in $G$ if there exists a subgroup chain $H=H₀≤ H₁≤… ≤ H_t=G$ such that either $H_i-1$ is normal in $H_i$ or $H_i/(H_i-1)_Hi$ is $σ$-primary for all $i=1,…,t$. A set $\mathcal{H}$ of subgroups of $G$ is said to be a \emph{complete Hall $σ$-set} of $G$ if every non-identity member of $\mathcal{H}$ is a Hall $σ_i$-subgroup of $G$ for some $i$ and $\mathcal{H}$ contains exactly one Hall $σ_i$-subgroup of $G$ for every $σ_i\in σ(G)$. Let $\mathcal{H}$ be a complete Hall $σ$-set of $G$. A subgroup $H$ of $G$ is said to be \emph{$\mathcal{H}$-permutable} if $HA=AH$ for all $A\in \mathcal {H}$. We say that a subgroup $H$ of $G$ is \emph{weakly $\mathcal {H}$-permutable} in $G$ if there exists a $σ$-subnormal subgroup $T$ of $G$ such that $G=HT$ and $H\cap T≤ H_{\mathcal {H}}$, where $H_{\mathcal {H}}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are $\mathcal {H}$-permutable. By using the weakly $\mathcal {H}$-permutable subgroups, we establish some new criteria for a group $G$ to be $σ$-soluble and supersoluble, and we also give the conditions under which a normal subgroup of $G$ is hypercyclically embedded.

ISO 690:

Cao, C., Amjid, V., Zhang, C. 2019. On weakly $\mathcal{H}$-permutable subgroups of finite groups. In Mathematica Slovaca, vol. 69, no.4, pp. 763-772. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0267

APA:

Cao, C., Amjid, V., Zhang, C. (2019). On weakly $\mathcal{H}$-permutable subgroups of finite groups. Mathematica Slovaca, 69(4), 763-772. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0267

Vydavateľ: Mathematical Institute, Slovak Academy of Sciences, Bratislava