Computations
Each search is a funnel: from all groups (or parents in the $p$-group tree) down to a few candidates whose $D_5$ is computed. The scale is logarithmic. The last step of every funnel is zero.
Order $2^{10}$, class $3$
Order $2^{11}$, rank $4$
§5, “Next”
Order $3^8$
Order $3^9$, ranks $3$–$4$
All runs
| What | Groups | Result | Time | Code |
|---|---|---|---|---|
| all $2$-groups of orders $8$–$128$ | 2 665 | $D_n=\gamma_n$ for all $n$ | compute | |
| groups of order $2^8$ of class $\ge3$ | 24 328 | $D_n=\gamma_n$ for all $n$ | compute | |
| metabelian groups of order $2^9$ of class $3$ (Jennings sieve) | 1 587 389 | $2792$ left by the sieve, all $D_4=D_5=1$ | compute | |
| order $2^{10}$, class $3$ (ANUPQ) | 3 400 | all candidates $D_4=D_5=1$ | $63$ + $21$ CPU-h | compute/pq1024 |
| order $2^{11}$, ranks $2$, $3$, $4$ | 53 171 | all candidates $D_4=D_5=1$ | $\approx20$ CPU-h (rank $4$) | compute/pq2048 |
| all $3$-groups of orders $3^4$–$3^6$ | 586 | $D_n=\gamma_n$ for all $n$ | compute | |
| order $3^7$, class $4$, after the sieve | 1 202 | $D_5=1$ by two methods | compute | |
| order $3^8$, non-metabelian of class $4$ | 699 | $z\notin D_5$ in all | $26$ min | compute/magnus |
| order $3^9$, ranks $3$–$4$ | 62 245 | $z\notin D_5$ in all; presentations rechecked with ANUPQ | $3$ h $21$ min on $4$ cores | compute/magnus |
| random groups with $Q/\gamma_2$ of exponent $3$, $3^{10}$–$3^{79}$ | 169 | $D_5=1$ in all (a check of Thm. 6.6) | $5$ min | compute/magnus |
| quotients of Rips-type groups, $2^{40}$–$2^{107}$ | 14 | $D_5=1$; $D_4\ne1$ in three | compute/magnus |
Method: for finite $G$ the lattice $\Delta^n$ is computed in Howell form over $\mathbb Z/p^E$; non-membership $g-1\notin\Delta^n+p^E\Delta$ proves $g\notin D_n$, membership is confirmed by the certificate $p^{E-1}\Delta\subseteq\Delta^n+p^E\Delta$. For large groups: the truncated Magnus algebra $\mathbb Z[G]/\Delta^n\cong T_{\lt n}(\mathbb Z^k)/I_n$ of a presentation on $k$ generators. Checks: two independent methods, PARI/GP, known values.