Geometric invariants for $p$-groups of class 2 and exponent $p$

We introduce geometric invariants for $p$-groups of class $2$ and exponent $p$. We report on their effectiveness in distinguishing among 5-generator $p$-groups of this type.

💡 Research Summary

The paper introduces a new family of geometric invariants for finite p‑groups of nilpotency class 2 and exponent p, and demonstrates their practical effectiveness in distinguishing such groups, especially those generated by five elements. The authors begin by recalling the Baer–MacLane correspondence, which associates to every class‑2, exponent‑p p‑group a skew‑symmetric matrix B(y) of linear forms over the field Fₚ. Here V = G/G′ (dimension n) and W = G′ (dimension d) are vector spaces, and the commutator map is encoded by B(y): V × V → W. The transpose (or adjoint) matrix B·(x) is defined analogously.

For each integer k, the k‑th determinantal ideal Iₖ(B) is generated by all k × k minors of B(y); similarly Jₗ(B·) is generated by the ℓ‑minors of the adjoint. The zero‑sets of these ideals in affine space, V_aff(Iₖ) and V_aff(Jₗ), are algebraic varieties whose geometric data (dimension, degree, number of Fₚ‑rational points, number of irreducible components, and the dimension of their linear span) are shown to be invariant under group isomorphism. Theorem 2.3 formalises this by listing four families of invariants that are preserved when two groups are isomorphic: (1) degrees and dimensions of the ideals, (2) degrees of the ideals appearing in a minimal primary decomposition, (3) degree, rational point count, and component count of the associated affine varieties, and (4) the dimension of the Fₚ‑span of each variety.

The authors then apply these invariants to concrete families of groups. First, they treat the six isomorphism classes of 4‑generator groups of order p⁷, class 2, exponent p, providing explicit matrices B₁,…,B₆. For each matrix they compute I₁(B) (which is zero, confirming that the derived subgroup lies in the centre) and the invariants of the adjoint’s I₃ and I₄. The resulting data (Table 1) already separate all six groups.

Next, the paper addresses the more intricate case of 5‑generator groups of order p⁸ (and, via immediate descendants, order p⁹). Prior work identifies 22 isomorphism classes. For each class the authors retrieve the corresponding matrix B and compute the invariants attached to I₄(B) and J₃(B·) for primes p = 3,…,37. Table 2 records, for each group, the number of Fₚ‑rational points of the relevant varieties, the degrees of the ideals, and the degrees of the components in a primary decomposition. Using only three of these invariants (the rational point count and two degree data) they uniquely identify 20 of the 22 groups; the remaining two (labelled 14 and 15) are indistinguishable by the presented invariants but have different automorphism‑group orders, which provides a secondary distinguishing criterion.

The final section connects the discussion to Higman’s PORC conjecture. Lee’s construction of a family L of groups of order p⁹ (for all p ≥ 5) yields groups whose automorphism‑group size varies in a non‑quasi‑polynomial way with p. The associated matrix B has I₄ generated by the square of a homogeneous quadratic f, so the projective variety V_proj(I₄) is a plane conic whose number of points depends on whether f is a square modulo p. Consequently, |Aut(G_B(Fₚ))| is not a quasi‑polynomial function of p. Theorem 4.1 shows that when n + d ≤ 7 the automorphism‑group size is always quasi‑polynomial, indicating that the pathological behaviour only appears for larger parameters.

Overall, the paper’s contributions are:

• Definition of a suite of geometric invariants derived from rank‑loci of skew‑symmetric linear‑form matrices.
• Proof that these invariants are preserved under group isomorphism.
• Empirical verification that they separate almost all 5‑generator class‑2, exponent‑p p‑groups up to order p⁸, with only a few borderline cases.
• Illustration of how these invariants interact with the PORC problem, providing concrete examples where quasi‑polynomial behaviour fails.
• Implementation notes indicating that the invariants can be computed in Magma via the MultilinearAlgebra package, and that the computations remain feasible for moderate primes.

The paper also acknowledges limitations: the current set of invariants does not resolve every pair (e.g., groups 14 and 15), and the computational cost of determinantal ideals grows quickly with larger d. Future work is suggested to explore additional tensor‑theoretic or geometric invariants and to optimise the algorithms for handling higher‑dimensional cases.