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On average orders of automorphism groups of bilinear maps over finite fields

Markus Bläser, Yinan Li, Youming Qiao, Alexander Rogovskyy

TL;DR

This work establishes sharp asymptotic bounds on the average order of automorphism groups for bilinear maps over finite fields, quantified by h(n,m,q) and its symmetric/alternating variants, in the regime m = Θ(n). The authors transform the problem into counting GL(V)×GL(W) orbits on tensor-type spaces via Burnside’s lemma, and analyze fixed points by decomposing matrices P,Q according to their rational normal forms, showing that most cases force P and Q to be almost scalar. The resulting bound h(n,m,q) ≤ q−1 + q^{−Ω(n^2)} yields near-tight counts for isomorphism classes of bilinear maps and has broad applications: asymptotically tight bounds for p-groups of Frattini class 2, probabilistic sampling of matrix spaces with trivial symmetry, and enumeration of cube-zero commutative algebras. These connections illustrate the central role of GL-orbit techniques in both classical group enumeration and modern linear-algebraic approaches to algebraic structures over finite fields, with implications for isomorphism testing and random-space constructions.

Abstract

Let $\varphi:V\times V\to W$ 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.

On average orders of automorphism groups of bilinear maps over finite fields

TL;DR

This work establishes sharp asymptotic bounds on the average order of automorphism groups for bilinear maps over finite fields, quantified by h(n,m,q) and its symmetric/alternating variants, in the regime m = Θ(n). The authors transform the problem into counting GL(V)×GL(W) orbits on tensor-type spaces via Burnside’s lemma, and analyze fixed points by decomposing matrices P,Q according to their rational normal forms, showing that most cases force P and Q to be almost scalar. The resulting bound h(n,m,q) ≤ q−1 + q^{−Ω(n^2)} yields near-tight counts for isomorphism classes of bilinear maps and has broad applications: asymptotically tight bounds for p-groups of Frattini class 2, probabilistic sampling of matrix spaces with trivial symmetry, and enumeration of cube-zero commutative algebras. These connections illustrate the central role of GL-orbit techniques in both classical group enumeration and modern linear-algebraic approaches to algebraic structures over finite fields, with implications for isomorphism testing and random-space constructions.

Abstract

Let be a bilinear map of finite vector spaces and over a finite field . We present asymptotic bounds on the number of isomorphism classes of bilinear maps under the natural action of and , when and are linearly related. As motivations and applications of the results, we present almost tight upper bounds on the number of -groups of Frattini class 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.

Paper Structure

This paper contains 29 sections, 25 theorems, 60 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Application 1: enumerating p-groups of Frattini class 2
  4. Application 2: random matrix spaces with trivial symmetry
  5. Application 3: enumerating cube-zero finite commutative algebras
  6. Organisation of the paper.
  7. Preliminary
  8. Notations.
  9. Bilinear maps and matrix spaces
  10. Conjugacy classes of glnq
  11. Average order of automorphism groups of bilinear maps
  12. Proof of Theorem \ref{['thm: tensors']}
  13. The setting.
  14. Turn to study fixed points.
  15. Partition according to conjugacy classes.
  16. ...and 14 more sections

Key Result

Theorem 1.1

For $C, R\in \mathbb{Q}$ with $C>0$ and $n\in \mathbb{N}$, let $m=m(n)=\lceil C\cdot n+R\rceil$. For large enough $n\in \mathbb{N}$, $h(n, m, q)$, $h_\mathrm{S}(n, m, q)$, and $h_\Lambda(n, m, q)$ are upper bounded by $q-1+\frac{1}{q^{\Omega(n^2)}}$.

Figures (2)

  • Figure 1: A Young diagram for the partition $(2, 2, 1, 1)$ (on the left) and its dual partition $(4,2)$ (on the right), which is obtained by transposing the diagram.
  • Figure 2: The rational normal form of $P$ with respect to the direct sum $U_1\oplus U_2$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2: Hig60, see Bla92 and BNV07
  • Theorem 1.3
  • Theorem 1.4: Wri71
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 2.1
  • Lemma 2.2: See e.g. Mac81
  • Definition 2.3: Rational normal forms
  • Remark 2.4
  • ...and 37 more