An explicit odd-order p-group with as many automorphisms as elements

Abstract

We construct a finite group G for which p = 1009, |G| = |Aut(G)| = 1009^52359. The group is the Baker–Campbell–Hausdorff group of an explicitly specified rank-31 Lie ring modulo 1009^1689. Its nilpotency class is at most 846 < 1009. An asymmetric lattice forces every automorphism of the finite Lie ring into an integral logarithm–exponential neighborhood. This gives a bijection between the full automorphism set and the kernel of a linear derivation matrix. Exact independently implemented Python and C++ computations establish the Smith valuations of that matrix. The resulting computer-assisted proof answers the existence question affirmatively. The construction and all exact certificates are supplied with separately implemented verification software.

The idea in plain English

An automorphism is a way to relabel every element of a group while preserving its multiplication. The question is whether a nontrivial group whose size is a power of an odd prime can have exactly as many of these symmetries as it has elements.

This paper constructs an example with 1009^52359 elements and the same number of automorphisms. The number is enormous, but the construction has a finite description: an element is a list of 31 numbers taken modulo 1009^1689, with a specified rule for multiplying two lists.

The main difficulty is accounting for every automorphism. The proof uses a deliberately asymmetric lattice to constrain them all, then turns their count into an exact linear-algebra calculation. The paper presents this as an affirmative answer to D. MacHale’s Kourovka Notebook problem 16.63.

The precise result

For p = 1009, the paper constructs a finite nontrivial p-group G with |G| = |Aut(G)| = 1009^52359, where Aut(G) is the full automorphism group.

The group is G = BCH(L/p^1689 L), where L is the explicit rank-31 Lie lattice in Definitions 2.1 and 5.1. Its nilpotency class is at most 846, below p, so the finite Lazard correspondence preserves its order and full automorphism group.

For this fixed Lie lattice and every integer i ≥ 7, the full Lie-ring automorphism group of L/p^i L has order p^(30i + 1689), while L/p^i L has p^(31i) elements. Choosing i = 1689 makes the two counts equal.

The exact derivation matrix has 14,415 rows, 961 columns, rank 931, and nullity 30. Its nonzero p-primary Smith valuations have multiplicities 87, 55, 758, 6, and 25 at valuations 0, 1, 2, 3, and 4; their sum is 1689.

Why the result holds

Begin with the rank-31 Lie-algebra template of Omirov and Ruan. A chosen flag has no nonidentity stabilizer in its full finite-field automorphism group. Weighting the lattice along this flag forces every automorphism of a sufficiently deep finite quotient into a domain where logarithms are integral.

Within that domain, logarithm and exponential preserve the required precision and give a bijection between all automorphisms and solutions of a linear derivation equation. This is the bridge from a nonlinear symmetry problem to an exact matrix count.

The matrix’s nullity contributes 30i to the exponent of the automorphism count, while its Smith valuations contribute 1689. The ring itself has order p^(31i), so the counts meet at i = 1689. The class bound then lets the Lazard correspondence transfer the equality to a finite group.

Scope and limitations

This gives one example at p = 1009. It does not establish an example for every odd prime, and it makes no minimality claim for the prime, group order, or number of generators.

The equality is between orders. The paper does not claim that G and Aut(G) are isomorphic.

The construction is explicit through its Lie bracket and finite Baker–Campbell–Hausdorff rule. It does not enumerate the group’s multiplication table or its automorphisms. Exact computation certifies the matrix data; the full-automorphism reduction and the correspondence with groups also require the written proof.

The Lie-algebra template is due to Omirov and Ruan. This paper supplies its integral normalization, finite-characteristic arguments, asymmetric lattice, and exact count of finite automorphisms.

The Kourovka Notebook lists this manuscript as an AI-proposed solution awaiting confirmation by mathematicians. Its listing should not be read as external validation or journal peer review.

Status and AI use

This is an unrefereed preprint whose construction, proof development, and initial verification software were generated using AI tools; subsequent checks are not external human peer review or proof-assistant formalization.

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Collaborate on this result

I welcome substantive collaboration on an independent review of the group-theoretic proof and exact certificates, particularly the reduction to the full automorphism count, or on preparing the work for journal submission.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). An explicit odd-order p-group with as many automorphisms as elements (Version 1.1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.22770864

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Background and prior work

• E. I. Khukhro and V. D. Mazurov (eds.), The Kourovka Notebook, 21st edition, October 2026 update-3: Problem 16.63, on printed page 106, asks the odd-prime equality question and links this manuscript with the marker for an AI-proposed solution awaiting mathematician confirmation. Archive 12.77 concerns the separate divisibility question.
• Bakhrom Omirov and Jie Ruan, On Lie Algebras with Only Inner Derivations (2026): Theorem 4.8 supplies the rank-31 characteristic-zero Lie-algebra template. The present construction credits that input and separately establishes the finite-characteristic facts it needs.
• Jon González-Sánchez and Andrei Jaikin-Zapirain, Finite p-groups with small automorphism group (2015): Constructs families with fewer automorphisms than elements and disproves the general divisibility conjecture. That result and its Lie-theoretic method are important background; it does not establish the exact equality sought in Problem 16.63.
• M. Shabani-Attar, On equality of order of a finite p-group and order of its automorphism group (2015): Classifies equality in specified classes and excludes odd-prime examples of maximal class, with cyclic Frattini subgroup, or of class two with cyclic centre. These restrictions are not a general nonexistence result.
• Serena Cicalò, Willem A. de Graaf, and Michael Vaughan-Lee, An effective version of the Lazard correspondence (2012): Explains the correspondence between finite p-groups and Lie rings of nilpotency class below p, including preservation of automorphism groups. That class restriction is checked explicitly here.