This is from MV2's open notebook on Kakeya-type problems (https://github.com/munimtechnologies/kakeya-small-fields). A finite 2-group whose involutions are all central: is it at most |Omega_1|^(e+1), where 2^e is its exponent? Below, one short lemma about a quadratic identity over F_2 reduces a large family of cases to anisotropic quadratic maps, gives M(3) = 3 for the number M(t) defined below, and answers the question with yes for every group with a self-centralising normal subgroup Z_4^a x Z_2^b with b <= 1 or a <= max(7, b + 2). References to other lemmas and theorems point to finite-groups.md in the repository, which also lists the exact computer checks; the ones for these results passed with every pre-registered prediction matched. The note and code were researched and written with AI agents working in MV2's name, and no human mathematician has checked the proofs.
Theorem 14 bounds a p-group by its elements of order p when p is odd. For p = 2 we ask:
Question. Let L be a finite 2-group of exponent 2^e whose involutions are all central, and let Omega = Omega_1(L), elementary abelian of rank k. Is |L| <= |Omega|^(e+1)?
Equality holds for Q_(2^n) (e = n - 1, k = 1), Q_8^m and Q_8^m x Z_4. The exponent e + 1 cannot be lowered, because of the generalised quaternion groups. A yes would give lambda*(G) >= 1/(e+1) in section 13. This section settles the question in several natural families. It does NOT settle the general case.
The rest of this section (new in 1.7.5) measures exactly how much codimension Lemma 20 needs, and shows that this one number decides the whole family. Let X be an F_2-space of dimension t and call a system a quadratic map sigma : V -> X with polar form beta, together with a linear map alpha : V -> End(X), written h -> A_h, satisfying the identity of Lemma 20,
A_(h') sigma(h) = (I + A_h) beta(h, h') for all h, h' in V. (*)
Let M(t) be the largest, over all systems, of the least codimension of a subspace of V on which sigma vanishes. By Lemma 21(a) below, M(t) <= t^2 + t, so it is finite.
Call a system anisotropic if sigma(h) != 0 for every h != 0. Then A_h is singular for every h outside K (Lemma 21(b)), restrictions to subspaces stay anisotropic, and the least codimension is dim V. So an anisotropic system with dim V = t + 1 would give M(t) > t.
Lemma 26 (restriction at a vector). Let (sigma, alpha) be a system, u in V, rho = rank A_u, X_1 = ker A_u and U = {h in V : beta(h, u) = 0}. (a) sigma(U) lies in X_1. Fix a complement of X_1 in X and, for h in U, let alpha_1(h) in End(X_1) be A_h restricted to X_1 followed by the projection onto X_1. Then sigma on U, with alpha_1, is a system with dim X_1 = t - rho. The same projection makes any subspace U' a system with values in any subspace X' that contains sigma(U'). (b) codim U <= t. If sigma(u) = 0, then codim U <= dim Fix(A_u) <= rho. (c) So if sigma(u) = 0, then LC(V) <= dim Fix(A_u) + M(t - rho) <= rho + M(t - rho). If A_u is invertible, sigma vanishes on U, so LC(V) <= t. Proof. (a) Put h' = u in (*): A_u sigma(h) = (I + A_h) beta(h, u), which is 0 for h in U. For U' and X' as in the last sentence, beta(h, h') = sigma(h + h') + sigma(h) + sigma(h') lies in X' for h, h' in U'. Let alpha'(h) be A_h restricted to X' followed by the projection onto X' along a fixed complement. Apply this projection to (*): the left side becomes alpha'(h') sigma(h), and, since beta(h, h') is in X', the right side becomes (I + alpha'(h)) beta(h, h'). The first claim is the case U' = U, X' = X_1. (b) U is the kernel of the linear map h -> beta(h, u) into X. If sigma(u) = 0, Lemma 25(a) puts its image into Fix(A_u), and Fix(A_u) n ker A_u = 0 gives dim Fix(A_u) <= t - dim ker A_u = rho. (c) A subspace of U on which sigma vanishes has codimension in V at most codim U plus its codimension in U. If A_u is invertible, X_1 = 0. QED
Let A(t) be the largest dim V of an anisotropic system with dim X = t. An anisotropic sigma is a t-tuple of quadratic forms with only the trivial common zero, so A(t) <= 2t by Chevalley-Warning, and a linear one-to-one sigma with alpha = 0 shows A(t) >= t.
Theorem 24 (M(t) is decided by anisotropic systems). For every t >= 1,
max(t, A(t)) <= M(t) <= t + max over 1 <= s <= t of max(A(s) - s, 0).
So M(t) <= 2t; M(t) = t for every t if and only if A(t) <= t for every t; and M(t) <= t + 1 for every t <= a if A(s) <= s + 1 for every s <= a. With A(1) = 1, A(2) = 2 and A(3) = 3 (Lemma 22 and Proposition 21), M(1) = 1, M(2) = 2 and M(3) = 3. Proof. The lower bound is Lemma 23(d) and the fact that an anisotropic system has LC = dim V. For the upper bound write E(t) for the maximum on the right and induct on t, with M(0) = 0. Given a system, pass to the reduced system on V/K_0 (Lemma 21(a)), which has the same least codimension. If it is anisotropic, LC = dim V <= A(t) <= t + E(t). Otherwise sigma has a zero u != 0, and A_u != 0, because a zero in K = ker alpha would lie in K_0. With rho = rank A_u >= 1, Lemma 26(c) and the induction hypothesis give LC <= rho + (t - rho) + E(t - rho) <= t + E(t). M(t) <= 2t follows from A(s) <= 2s. A(1) <= M(1) = 1 is Lemma 22. QED (for t = 2 and 3, computer-assisted through Proposition 21.)
So every system with t = 3 has a vanishing subspace of codimension <= 3, which is sharp by Lemma 23(d), and Theorem 23 is superseded. It stays true, and m3_core.py still checks it.
Proposition 22 (anisotropic systems). Let (sigma, alpha) be anisotropic with dim X = t. Then every A_h is singular, and: (a) if some A_u has rank rho, then dim V <= t + A(t - rho); (b) if every A_h has rank <= 1, then either all A_h have one common image line, and dim V <= t, or all have one common kernel, and dim V <= 2 + A(t - 1); (c) hence A(t) <= max(t, 2 + A(t - 1), t + A(t - 2)) for t >= 2, so A(4) <= 6 and A(5) <= 8. Also A(t_1 + t_2) >= A(t_1) + A(t_2), and A is non-decreasing. By Theorem 24, M(4) <= 4 + 2 = 6 and M(5) <= 5 + 3 = 8. Proof. A_h sigma(h) = 0 with sigma(h) != 0 makes A_h singular. (a) By Lemma 26(a), U carries an anisotropic system with dim X_1 = t - rho, and codim U <= t. (b) In a linear space of matrices of rank <= 1, any two non-zero elements x f and y g share the image (x = y) or the kernel (f = g), since otherwise x f + y g has rank 2; and if x f and x g with f != g are present, a third element y f' with y != x would need f' = f and f' = g. So one image or one kernel is common to all. Common image span(x_0): A_h = x_0 f_h with f_h linear in h, and (*) reads beta(h, h') = x_0 (f_(h')(sigma(h)) + f_h(beta(h, h'))), so beta takes values in span(x_0). Then sigma modulo x_0 is additive, so linear, and on its kernel U' (codimension <= t - 1) sigma takes values in span(x_0). By Lemma 26(a), U' is an anisotropic system with t = 1, so dim U' <= A(1) = 1. Common kernel ker f_0: A_h = g_h f_0 with g_h linear in h. Apply f_0 to (*): with q = f_0(sigma), B = f_0(beta) and p(h) = f_0(g_h), q(h) p(h') = B(h, h') (1 + p(h)), which is Lemma 15 with a = 1. So q vanishes on a subspace R of codimension <= 2 (rad B if B != 0; ker q if B = 0; all of V if q = 0), and on R sigma takes values in ker f_0. By Lemma 26(a), R is an anisotropic system with t - 1, so dim R <= A(t - 1). (c) If alpha = 0, sigma is linear and one-to-one, so dim V <= t. If some A_h has rank >= 2, use (a) with A(t - rho) <= A(t - 2). Otherwise use (b). For the sum, put two systems side by side (block-diagonal A_h, sigma = (sigma_1, sigma_2)); adding a coordinate on which sigma is 0 shows that A is non-decreasing. QED
Corollary 14 (a <= 7, and a <= b + 2). If L has a self-centralising normal subgroup A ~ Z_4^a x Z_2^b with b <= 1 or a <= max(7, b + 2), then |L| <= |Omega|^4, and the answer is yes. This contains Corollary 13 and settles, for example, (6, 2), (5, 3), (7, 2), (6, 3), (7, 5) and (10, 8). The first open cases are (8, b) for 2 <= b <= 5 and (b + 3, b) for b >= 6. Proof. b <= 1 is Theorems 16 and 17, so let b >= 2; if S = 0, then e <= 2. By Theorem 24, M(t) - t <= E(t), where E(1) = E(2) = E(3) = 0, E(4) <= 2 and E(5) <= 3 by Proposition 22, and E(t) <= t always. Let tau be the largest t <= a with E(s) <= b for every s <= t. So tau >= min(a, 4) when b = 2, tau >= min(a, 5) when 3 <= b <= 5, and tau >= min(a, b) when b >= 6. If some y != 0 has dim(V_y n S) = t <= tau, Proposition 20 gives n <= 2a + 2b + M(t) - t <= 2a + 3b. This happens when dim S <= tau, and, by Lemma 19, when codim W <= (tau + 1) b - 1, that is r >= b(a - tau - 1) + 1. Otherwise dim S >= tau + 1 and r <= min(b(a - tau - 1), b(a - s)), and Theorem 17 gives n <= a + s + 2b + r <= a + 2b + ab - (tau + 1)(b - 1) (use the first bound on r when s <= tau + 1 and the second when s >= tau + 1). This is <= 2a + 3b exactly when (a - tau - 2)(b - 1) <= 1: for b = 2 when a <= tau + 3, and for b >= 3 when a <= tau + 2. With the values of tau above this is a <= 7 for 2 <= b <= 5 and a <= b + 2 for b >= 6. QED (computer-assisted through Proposition 21, via Theorem 24 and Proposition 22.)
For the first open case (8, 2), the same count needs tau >= 5, that is E(5) <= 2, which holds if A(5) <= 7. For (8, b) with 3 <= b <= 5 it needs tau >= 6, which holds if A(6) <= 6 + b. Both would follow from A(t) = t, which is true for t <= 3.
Version 1.7.8 of the file adds Lemmas 27 and 28 and Corollary 15: at the least t where A(t) = t fails, a counterexample needs an irreducible space of singular matrices, and the open case (8, 2) needs only the irreducible spaces of matrices of rank at most 2 in M_5(F_2).
Prior work. Searches on 2026-10-15 and 2026-10-16 found no statement of this question. The literature's "2-central" means Omega_2 <= Z, which is a stronger hypothesis. Higman's Suzuki 2-groups (1963) are the case e = 2 in a special setting. In section 2 of arXiv math/0009003, Bovdi and Dokuchaev consider a non-abelian 2-group G of exponent 4 whose involutions are central, with Phi(G) = Omega(G) of order 4, and cite a result of N. Blackburn (Huppert and Blackburn, Finite Groups II, Theorem VIII.5.4) for |G| <= |Omega(G)|^3. That is our bound for e = 2 in their setting, so the case e = 2 may well be known more generally; we have not read Blackburn's theorem. A search on 2026-10-17 found Pakianathan and Yalcin, "Quadratic maps and Bockstein closed group extensions" (arXiv math/0606374), on central extensions of elementary abelian 2-groups through their quadratic maps, including the map A -> A + A^2 on gl_n(F_2). From its abstract and a search summary of its text (we have not read it in full), it appears to record the Chevalley-Warning bound for quadratic maps with no non-trivial zero, with equality for Q_8^m, which is the argument of Lemma 10. We found nothing on the identity (*) or on M(t). The tools (quadratic maps, Chevalley-Warning, determinants of degree-bounded matrices, spaces of rank-1 matrices, invariant subspaces) are standard, so a specialist may know these bounds. Please open an issue if so.
Licence: Text CC BY 4.0; code Apache-2.0 (see LICENSE). By MV2 of Munim, Inc.; researched and written with AI agents working in his name.
Top comments (0)