On generalized covering and avoidance properties of finite groups and saturated fusion systems
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| Format: | Preprint |
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2023
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| _version_ | 1866916510380851200 |
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| author | Zhang, Shengmin Shen, Zhencai |
| author_facet | Zhang, Shengmin Shen, Zhencai |
| contents | A subgroup $A$ of a finite group $G$ is said to be a $CAP$-subgroup of $G$, if for any chief factor $H/K$ of $G$, either $A H= AK$ or $A\cap H = A \cap K$. Let $p$ be a prime, $S$ be a $p$-group and $\mathcal{F}$ be a saturated fusion system over $S$. Then $\mathcal{F}$ is said to be supersolvable, if there exists a series of $S$, namely $1 = S_0 \leq S_1 \leq \cdots \leq S_n = S$, such that $S_{i+1}/S_i$ is cyclic, and $S_i$ is strongly $\mathcal{F}$-closed for any $i=0,1,\cdots,n$. In this paper, we first introduce the concept of strong $p$-$CAP$-subgroups, and investigate the structure of finite groups under the assumptions that some subgroups of $G$ are partial $CAP$-subgroups or strong $(p)$-$CAP$-subgroups of $G$, and obtain some criteria for a group $G$ to be $p$-supersolvable. After that, we investigate the characterizations for supersolvability of $\mathcal{F}_S (G)$ under the assumptions that some subgroups of $G$ are partial $CAP$-subgroups or strong $(p)$-$CAP$-subgroups of $G$, and obtain some criteria for a fusion system $\mathcal{F}_S (G)$ to be supersolvable. The above results improve some known results and develop some new results about $CAP$-subgroups from fusion systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_00012 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On generalized covering and avoidance properties of finite groups and saturated fusion systems Zhang, Shengmin Shen, Zhencai Group Theory 20D10, 20D15, 20D20 A subgroup $A$ of a finite group $G$ is said to be a $CAP$-subgroup of $G$, if for any chief factor $H/K$ of $G$, either $A H= AK$ or $A\cap H = A \cap K$. Let $p$ be a prime, $S$ be a $p$-group and $\mathcal{F}$ be a saturated fusion system over $S$. Then $\mathcal{F}$ is said to be supersolvable, if there exists a series of $S$, namely $1 = S_0 \leq S_1 \leq \cdots \leq S_n = S$, such that $S_{i+1}/S_i$ is cyclic, and $S_i$ is strongly $\mathcal{F}$-closed for any $i=0,1,\cdots,n$. In this paper, we first introduce the concept of strong $p$-$CAP$-subgroups, and investigate the structure of finite groups under the assumptions that some subgroups of $G$ are partial $CAP$-subgroups or strong $(p)$-$CAP$-subgroups of $G$, and obtain some criteria for a group $G$ to be $p$-supersolvable. After that, we investigate the characterizations for supersolvability of $\mathcal{F}_S (G)$ under the assumptions that some subgroups of $G$ are partial $CAP$-subgroups or strong $(p)$-$CAP$-subgroups of $G$, and obtain some criteria for a fusion system $\mathcal{F}_S (G)$ to be supersolvable. The above results improve some known results and develop some new results about $CAP$-subgroups from fusion systems. |
| title | On generalized covering and avoidance properties of finite groups and saturated fusion systems |
| topic | Group Theory 20D10, 20D15, 20D20 |
| url | https://arxiv.org/abs/2402.00012 |