Absolutely irreducible quasisimple linear groups containing elements of order a specified Zsigmondy prime
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915016637153280 |
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| author | Glasby, S. P. Niemeyer, Alice C. Praeger, Cheryl E. Zalesski, A. E. |
| author_facet | Glasby, S. P. Niemeyer, Alice C. Praeger, Cheryl E. Zalesski, A. E. |
| contents | This paper is concerned with absolutely irreducible quasisimple subgroups $G$ of a finite general linear group $GL_d(\mathbb{F}_q)$ for which some element $g\in G$ of prime order $r$, in its action on the natural module $V=(\mathbb{F}_q)^d$, is irreducible on a subspace of the form $V(1-g)$ of dimension $d/2$. We classify $G,d,r$, the characteristic $p$ of the field $\mathbb{F}_q$, and we identify those examples where the element $g$ has a fixed point subspace of dimension $d/2$. Our proof relies on representation theory, in particular, the multiplicities of eigenvalues of $g$, and builds on earlier results of DiMuro. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08270 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Absolutely irreducible quasisimple linear groups containing elements of order a specified Zsigmondy prime Glasby, S. P. Niemeyer, Alice C. Praeger, Cheryl E. Zalesski, A. E. Representation Theory Group Theory 20C20, 20G05, 20H20 This paper is concerned with absolutely irreducible quasisimple subgroups $G$ of a finite general linear group $GL_d(\mathbb{F}_q)$ for which some element $g\in G$ of prime order $r$, in its action on the natural module $V=(\mathbb{F}_q)^d$, is irreducible on a subspace of the form $V(1-g)$ of dimension $d/2$. We classify $G,d,r$, the characteristic $p$ of the field $\mathbb{F}_q$, and we identify those examples where the element $g$ has a fixed point subspace of dimension $d/2$. Our proof relies on representation theory, in particular, the multiplicities of eigenvalues of $g$, and builds on earlier results of DiMuro. |
| title | Absolutely irreducible quasisimple linear groups containing elements of order a specified Zsigmondy prime |
| topic | Representation Theory Group Theory 20C20, 20G05, 20H20 |
| url | https://arxiv.org/abs/2411.08270 |