Kronecker classes, normal coverings and chief factors of groups
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| Format: | Preprint |
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2024
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| _version_ | 1866909989172412416 |
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| author | Fusari, Marco Harper, Scott Spiga, Pablo |
| author_facet | Fusari, Marco Harper, Scott Spiga, Pablo |
| contents | For a group $G$, a subgroup $U \leq G$ and a group $\mathrm{Inn}(G) \leq A \leq \mathrm{Aut}(G)$, we say that $U$ is an $A$-covering group of $G$ if $G = \bigcup_{a\in A}U^a$. A theorem of Jordan (1872) implies that if $G$ is a finite group, $A = \mathrm{Inn}(G)$ and $U$ is an $A$-covering group of $G$, then $U = G$. Motivated by a question concerning Kronecker classes of field extensions, Neumann and Praeger (1988) conjectured that, more generally, there is an integer function $f$ such that if $G$ is a finite group and $U$ is an $A$-covering subgroup of $G$, then $|G:U| \leq f(|A:\mathrm{Inn}(G)|)$. A key piece of evidence for this conjecture is a theorem of Praeger (1994), which asserts that there is a two-variable integer function $g$ such that if $G$ is a finite group and $U$ is an $A$-covering subgroup of $G$, then $|G:U|\leq g(|A:\mathrm{Inn}(G)|,c)$ where $c$ is the number of $A$-chief factors of~$G$. Unfortunately, the proof of this result contains an error. In this paper, using a different argument, we give a correct proof of this theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_02569 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kronecker classes, normal coverings and chief factors of groups Fusari, Marco Harper, Scott Spiga, Pablo Group Theory Combinatorics For a group $G$, a subgroup $U \leq G$ and a group $\mathrm{Inn}(G) \leq A \leq \mathrm{Aut}(G)$, we say that $U$ is an $A$-covering group of $G$ if $G = \bigcup_{a\in A}U^a$. A theorem of Jordan (1872) implies that if $G$ is a finite group, $A = \mathrm{Inn}(G)$ and $U$ is an $A$-covering group of $G$, then $U = G$. Motivated by a question concerning Kronecker classes of field extensions, Neumann and Praeger (1988) conjectured that, more generally, there is an integer function $f$ such that if $G$ is a finite group and $U$ is an $A$-covering subgroup of $G$, then $|G:U| \leq f(|A:\mathrm{Inn}(G)|)$. A key piece of evidence for this conjecture is a theorem of Praeger (1994), which asserts that there is a two-variable integer function $g$ such that if $G$ is a finite group and $U$ is an $A$-covering subgroup of $G$, then $|G:U|\leq g(|A:\mathrm{Inn}(G)|,c)$ where $c$ is the number of $A$-chief factors of~$G$. Unfortunately, the proof of this result contains an error. In this paper, using a different argument, we give a correct proof of this theorem. |
| title | Kronecker classes, normal coverings and chief factors of groups |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/2410.02569 |