Tactical decompositions in finite polar spaces and non-spreading classical group actions
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929398368698368 |
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| author | Bamberg, John Giudici, Michael Lansdown, Jesse Royle, Gordon F. |
| author_facet | Bamberg, John Giudici, Michael Lansdown, Jesse Royle, Gordon F. |
| contents | For finite classical groups acting naturally on the set of points of their ambient polar spaces, the symmetry properties of \emph{synchronising} and \emph{separating} are equivalent to natural and well-studied problems on the existence of certain configurations in finite geometry. The more general class of \emph{spreading} permutation groups is harder to describe, and it is the purpose of this paper to explore this property for finite classical groups. In particular, we show that for most finite classical groups, their natural action on the points of its polar space is non-spreading. We develop and use a result on tactical decompositions (an \emph{AB-Lemma}) that provides a useful technique for finding witnesses for non-spreading permutation groups. We also consider some of the other primitive actions of the classical groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_17576 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tactical decompositions in finite polar spaces and non-spreading classical group actions Bamberg, John Giudici, Michael Lansdown, Jesse Royle, Gordon F. Group Theory Combinatorics 05E18, 20B15, 51E20, 51E21 For finite classical groups acting naturally on the set of points of their ambient polar spaces, the symmetry properties of \emph{synchronising} and \emph{separating} are equivalent to natural and well-studied problems on the existence of certain configurations in finite geometry. The more general class of \emph{spreading} permutation groups is harder to describe, and it is the purpose of this paper to explore this property for finite classical groups. In particular, we show that for most finite classical groups, their natural action on the points of its polar space is non-spreading. We develop and use a result on tactical decompositions (an \emph{AB-Lemma}) that provides a useful technique for finding witnesses for non-spreading permutation groups. We also consider some of the other primitive actions of the classical groups. |
| title | Tactical decompositions in finite polar spaces and non-spreading classical group actions |
| topic | Group Theory Combinatorics 05E18, 20B15, 51E20, 51E21 |
| url | https://arxiv.org/abs/2403.17576 |