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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2509.21139 |
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| _version_ | 1866911614369792000 |
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| author | Villareal, Jonathon |
| author_facet | Villareal, Jonathon |
| contents | Let $\mathcal{L}$ be a centric linking system associated to a saturated fusion system on a finite $p$-group $S$. An automorphism of $\mathcal{L}$ is said to be rigid if it restricts to the identity on the fusion system. An inner rigid automorphism is conjugation by some element of the center of $S$. If $\mathcal{L}$ is the centric linking system of a finite group $G$, then rigid automorphisms of $\mathcal{L}$ are closely related to automorphisms of $G$ that centralize $S$. For odd primes, all rigid automorphisms are known to be inner, but this fails for the prime 2. We determine which known quasisimple linking systems at the prime 2 have a noninner rigid automorphism. Based on previous results, this reduces to handling the case of the linking systems at the prime 2 of finite simple groups of Lie type in odd characteristic. These have no noninner rigid automorphisms with two families of exceptions: the 2-dimensional projective special linear groups and even-dimensional orthogonal groups for quadratic forms of nonsquare discriminant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_21139 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigid automorphisms of linking systems of finite groups of Lie type Villareal, Jonathon Group Theory Let $\mathcal{L}$ be a centric linking system associated to a saturated fusion system on a finite $p$-group $S$. An automorphism of $\mathcal{L}$ is said to be rigid if it restricts to the identity on the fusion system. An inner rigid automorphism is conjugation by some element of the center of $S$. If $\mathcal{L}$ is the centric linking system of a finite group $G$, then rigid automorphisms of $\mathcal{L}$ are closely related to automorphisms of $G$ that centralize $S$. For odd primes, all rigid automorphisms are known to be inner, but this fails for the prime 2. We determine which known quasisimple linking systems at the prime 2 have a noninner rigid automorphism. Based on previous results, this reduces to handling the case of the linking systems at the prime 2 of finite simple groups of Lie type in odd characteristic. These have no noninner rigid automorphisms with two families of exceptions: the 2-dimensional projective special linear groups and even-dimensional orthogonal groups for quadratic forms of nonsquare discriminant. |
| title | Rigid automorphisms of linking systems of finite groups of Lie type |
| topic | Group Theory |
| url | https://arxiv.org/abs/2509.21139 |