[Paper Review] The automorphisms of class two groups of prime exponent

Michael Vaughan-Lee|arXiv (Cornell University)|Jan 4, 2015

Finite Group Theory Research5 references3 citations

TL;DR

This paper classifies all class two $p$-groups of prime exponent up to order $p^8$, proving that for each such group, both the number of conjugacy classes and the order of the automorphism group are polynomial in $p$. It identifies the minimal counterexample—$G_p$ of order $p^9$—where both invariants fail to be PORC (Polynomial on Residue Classes), demonstrating that the PORC conjecture for $f(p^n)$ may fail at $n=10$. The result hinges on explicit computation of automorphism group actions and conjugacy class counts via elliptic curve and group extension techniques.

ABSTRACT

We give a complete list of all the 70 class two groups of exponent p (p>2) and order p^k for k<9. For each of these groups the number of conjugacy classes is a polynomial in p, and the order of the automorphism group is a polynomial in p. In contrast, Marcus du Sautoy and Michael Vaughan-Lee have given an example of a class two group of exponent p and order p^9 for which neither the number of conjugacy classes, nor the order of the automorphism group is polynomial on residue classes (PORC).

Motivation & Objective

• To systematically classify all class two $p$-groups of prime exponent with order $p^n$ for $n \leq 8$.
• To determine whether the number of conjugacy classes and the order of the automorphism group are PORC functions of $p$ for these groups.
• To identify the smallest group where these invariants fail to be PORC, thereby testing the limits of Higman's PORC conjecture.
• To provide explicit structural and computational descriptions of automorphism groups and conjugacy class counts for small $p$-groups.

Proposed method

• Enumerated all isomorphism classes of class two $p$-groups of exponent $p$ and order $p^n$ for $n \leq 8$, resulting in 70 such groups.
• Used group presentations and commutator relations to define the structure of each group, particularly focusing on the derived subgroup and central extensions.
• Computed the number of conjugacy classes via group-theoretic formulas involving the size of orbits under automorphism actions and trace identities.
• Determined the automorphism group order by analyzing the action on the Frattini quotient $G/G'$, using linear algebra over $\mathrm{GF}(p)$.
• Employed elliptic curve point counts (e.g., $y^2 = x^3 - x$) to express conjugacy class counts in non-PORC cases, particularly for $G_p$.
• Applied the $p$-group generation algorithm to analyze descendants of $G_p$, linking automorphism group structure to the number of non-PORC descendants.

Experimental results

Research questions

• RQ1 Are the number of conjugacy classes and the order of the automorphism group polynomial in $p$ for all class two $p$-groups of exponent $p$ and order $p^n$ with $n \leq 8$?
• RQ2 Does the failure of the PORC property for $f(p^n)$ occur at $n=10$, and if so, what is the minimal group causing this?
• RQ3 Can the automorphism group of a $p$-group be computed explicitly using linear algebra over $\mathrm{GF}(p)$, especially when the group has a nontrivial commutator structure?
• RQ4 How does the number of solutions to elliptic curve equations over $\mathrm{GF}(p)$ influence the conjugacy class count in certain $p$-groups?
• RQ5 Is there a minimal $p$-group of class two and prime exponent where both the number of conjugacy classes and automorphism group order fail to be PORC functions of $p$?

Key findings

• For all 70 class two $p$-groups of exponent $p$ and order $p^n$ with $n \leq 8$, the number of conjugacy classes is polynomial in $p$.
• The order of the automorphism group is polynomial in $p$ for all such groups, including the group with 7 generators and 3 commuting pairs.
• The group $G_p$ of order $p^9$ is the smallest known example where both the number of conjugacy classes and the automorphism group order are not PORC functions of $p$.
• The number of descendants of $G_p$ of order $p^{10}$ and exponent $p$ is not PORC, with its value depending on $p \mod 12$ and the number of points on the elliptic curve $y^2 = x^3 - x$ over $\mathrm{GF}(p)$.
• The automorphism group order of $G_p$ is not PORC: it is $|\mathrm{GL}(2,p)| \cdot 4p^{18}$ for $p \equiv 5 \mod 12$, $|\mathrm{GL}(2,p)| \cdot 2p^{18}$ for $p \equiv 7 \mod 12$, and so on, with distinct formulas depending on $p \mod 12$ and the non-vanishing of $V_p$.
• The conjugacy class count for $G_p$ is $p^6 + p^3 - 1 + (p^3 - p^2 - p + 1) \cdot E$, where $E$ is the number of points on $y^2 = x^3 - x$ over $\mathrm{GF}(p)$, and $E$ is not a PORC function.