On average orders of automorphism groups of bilinear maps over finite fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909532687433728 |
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| author | Bläser, Markus Li, Yinan Qiao, Youming Rogovskyy, Alexander |
| author_facet | Bläser, Markus Li, Yinan Qiao, Youming Rogovskyy, Alexander |
| contents | Let $φ:V\times V\to W$ be a bilinear map of finite vector spaces $V$ and $W$ over a finite field $\mathbb{F}_q$. We present asymptotic bounds on the number of isomorphism classes of bilinear maps under the natural action of $\mathrm{GL}(V)$ and $\mathrm{GL}(W)$, when $\dim(V)$ and $\dim(W)$ are linearly related.
As motivations and applications of the results, we present almost tight upper bounds on the number of $p$-groups of Frattini class $2$ as first studied by Higman (Proc. Lond. Math. Soc., 1960). Such bounds lead to answers for some open questions by Blackburn, Neumann, and Venkataraman (Cambridge Tracts in Mathematics, 2007). Further applications include sampling matrix spaces with the trivial automorphism group, and asymptotic bounds on the number of isomorphism classes of finite cube-zero commutative algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07299 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On average orders of automorphism groups of bilinear maps over finite fields Bläser, Markus Li, Yinan Qiao, Youming Rogovskyy, Alexander Combinatorics Commutative Algebra Group Theory Let $φ:V\times V\to W$ be a bilinear map of finite vector spaces $V$ and $W$ over a finite field $\mathbb{F}_q$. We present asymptotic bounds on the number of isomorphism classes of bilinear maps under the natural action of $\mathrm{GL}(V)$ and $\mathrm{GL}(W)$, when $\dim(V)$ and $\dim(W)$ are linearly related. As motivations and applications of the results, we present almost tight upper bounds on the number of $p$-groups of Frattini class $2$ as first studied by Higman (Proc. Lond. Math. Soc., 1960). Such bounds lead to answers for some open questions by Blackburn, Neumann, and Venkataraman (Cambridge Tracts in Mathematics, 2007). Further applications include sampling matrix spaces with the trivial automorphism group, and asymptotic bounds on the number of isomorphism classes of finite cube-zero commutative algebras. |
| title | On average orders of automorphism groups of bilinear maps over finite fields |
| topic | Combinatorics Commutative Algebra Group Theory |
| url | https://arxiv.org/abs/2503.07299 |