Derived Subgroups, Automorphisms, and Solvable Group-Algebra Units

Definitions and mathematical statement

For a group $G$, use the derived series $G^{(0)}=G$, $G^{(n+1)}=[G^{(n)},G^{(n)}]$. The first question asks, for every free group $G$ and every intended index $n$, whether \[ G\cap\left(1+\delta_n({\mathbb{Z}} G)\right)=G^{(n)}. \] The intersection uses the standard embedding $G\subseteq({\mathbb{Z}} G)^\times$. The source does not define $\delta_n$. Its derived-series context suggests a Lie-derived construction in the group ring, but the additive Lie-derived series, the associative ideals it generates, and augmentation-ideal powers are different constructions and cannot be interchanged. The question also requests a description of the analogous intersection over fields of positive characteristic and for arbitrary groups; no particular general formula or equality is supplied for those extensions. The index origin and the precise $\delta_n$ convention must be recovered before the displayed equality is a self-contained assertion.

Second, let $G$ be a finite $p$-group and $k={\mathbb{F}}_p$. Every $\sigma\in\operatorname{Aut}(G)$ extends linearly to a $k$-algebra automorphism $\widehat\sigma$ of $kG$. The requested factorization is \[ \forall\alpha\in\operatorname{Aut}_{k\text{-alg}}(kG),\qquad \exists\sigma\in\operatorname{Aut}(G)\ \exists\psi\text{ unipotent}: \alpha=\psi\circ\widehat\sigma. \] This order expresses an automorphism induced by $G$ followed by a unipotent automorphism. The record does not define unipotent. In the ordinary linear meaning it would require $(\psi-\operatorname{id})^m=0$ on the finite-dimensional vector space $kG$ for some $m$; a convention defined through the augmentation filtration would need to be specified separately.

Third, for a field $K$ of odd positive characteristic, regard $A=KG$ as a Lie algebra with bracket $[a,b]=ab-ba$. Put $L^{(0)}=A$ and $L^{(m+1)}=\operatorname{span}_K\{[a,b]:a,b\in L^{(m)}\}$. Its Lie-derived length $\operatorname{dl}^L(A)$ is the least $m$ with $L^{(m)}=0$. For a group $B$, $\operatorname{dl}(B)$ is the least $m$ with $B^{(m)}=1$, or infinity if there is none. The question is whether a single function $f:{\mathbb{N}}\to{\mathbb{N}}$, independent of the field, group, and characteristic, satisfies \[ \operatorname{dl}(U(KG))\leq f\left(\operatorname{dl}^L(KG)\right) \] for every Lie-solvable $KG$ in the stated odd-characteristic class. Here $U(KG)=(KG)^\times$ is the full unit group. No finite-group hypothesis is supplied in this third part, no numerical form of $f$ is prescribed, and characteristic two is excluded.

Short English statement

The record asks three separate questions about a derived-series intersection in a group ring, a factorization of automorphisms over the prime field, and a universal bound for the derived length of units in odd characteristic. The first two require missing definitions of the ring filtration and of unipotence.

Sources

• Local UnsolvedMath registry, permanent ID $30000887$, exact source code catalogue entry, proposed year 2007; no individual proposer is supplied.
• Mini-Workshop: Arithmetik von Gruppenringen (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/55, pp. 3209--3240. https://doi.org/10.4171/owr/2007/55.
• Supplied reference labelled "derived length of group-algebra units": https://doi.org/10.21136/CMJ.2017.0205-16.
• Supplied reference labelled "automorphisms of modular group algebras": https://doi.org/10.1016/j.jalgebra.2023.08.019.
• The supplied corrected_verified note says all three questions and their notation were retained. The assessment checked 2026-08-21 reports important-family results for dimension-subgroup and modular-automorphism cases and an unresolved general derived-length bound. The partially solved label is dated supplied triage requiring independent verification of exact theorem scope.