Open questions

What is left, most important first.

1. Problem 20.84 itself: is $D_5(Q)=1$ for all finite $2$-groups of class $3$? A counterexample must have $D_4(Q)\ne1$ and order $\ge2^{11}$.

2. Order $2^{11}$, ranks $5$–$7$. Rank $5$ is the bulk of the cost (about $100$ CPU-days by estimate); at rank $7$ the stock ANUPQ overflows.

3. Is there $3$-torsion in $D_5/\gamma_5$, i.e. is Tahara's bound $6$ attained? A minimal example has order $\ge3^{10}$, class $4$, cyclic centre and at least two invariants $\ge9$ in $Q/\gamma_2$.

4. The next case: $Q/\gamma_2$ with two invariants $\ge9$, e.g. $C_9\times C_9\times C_3$. There $\gamma_2/\gamma_3$ may have exponent $9$, and the proof of Theorem 6.8 does not go through.

5. An exact criterion for nonterminality of groups of type $(d,2,1)$: at $3^8$, $190$ of the $7\,824$ have descendants, and the known necessary conditions also hold for $573$ terminal ones.

6. The group analogue of Theorem C: $D_{2c-1}(G)=1$ for metabelian groups of class $\le c$? Open for $p=2$, $c\ge3$ and for odd $p$, $c\ge p+2$.

7. Central products glued along a proper subgroup when $\sigma$ is not a $2^{j+1}$-th power: $12$ of the $212$ splittings of candidates of orders $2^8$–$2^9$ are covered by nothing. A projection to $H$ cannot do it in general: for $D_8\circ_{\langle t\rangle}M_{16}$ the ring equality fails at $n=4,5$.