Isomorphism and Torsion-Unit Problems for Group Algebras

Definitions and mathematical statement

The record combines three separate questions; their groups and coefficient fields need not be the same.

First, let $G$ be a finite $p$-group. The source assumes that $G$, embedded as group elements in $(kG)^\times$, is a "complement factor" in that unit group, and asks \[ kG\cong kH\text{ as }k\text{-algebras}\quad\Longrightarrow\quad G\cong H\ ? \] The field $k$, its characteristic, the group class for $H$, and the complement-factor condition are not defined. A normal-complement condition would mean $(kG)^\times=NG$, $N\triangleleft(kG)^\times$, $N\cap G=1$; a direct-factor condition would additionally require commuting factors. Neither interpretation is silently selected, nor is the full unit group replaced by its augmentation-one subgroup.

Second, take $G=\operatorname{GL}_n(k)$, where $k$ is a finite field and $n\geq1$. For $u=\sum_{g\in G}a_g g\in{\mathbb{Z}} G$, put \[ \varepsilon(u)=\sum_ga_g,\qquad V({\mathbb{Z}} G)=\{u\in({\mathbb{Z}} G)^\times:\varepsilon(u)=1\},\qquad \varepsilon_C(u)=\sum_{g\in C}a_g \] for each conjugacy class $C$ of $G$. A torsion unit has finite multiplicative order. The question is \[ \forall u\in V({\mathbb{Z}} G)\text{ of finite order},\qquad \varepsilon_{\pi(u)}(u)\ne0\ ? \] The source does not define $\pi$, or whether its subscript denotes an element's conjugacy class. A natural candidate is the map induced by ${\mathbb{Z}} G\to\operatorname{Mat}_n(k)$, sending each group element to its matrix and integer coefficients to $k$; it sends units to $\operatorname{GL}_n(k)$. If that is the intended map, the subscript means the conjugacy class of this matrix. The candidate is identified only to expose the missing convention, not asserted as a recovered definition.

Third, let $(u_p)_{p\in\mathcal P}$ be the nonempty family of torsion units constructed in the report from central primitive idempotents of localized cyclotomic group algebras, viewed in ${\mathbb{C}} G$. The construction, local rings, and index set $\mathcal P$ are absent. The question is whether \[ [\ \forall p,q\in\mathcal P\ \exists x_{p,q}\in({\mathbb{C}} G)^\times: x_{p,q}^{-1}u_px_{p,q}=u_q\ ] \] \[ \Longrightarrow\quad [\ \exists g\in G\ \forall p\in\mathcal P\ \exists y_p\in({\mathbb{C}} G)^\times: y_p^{-1}u_py_p=g\ ]. \] By mutual conjugacy, being conjugate to a group element for one index would imply it for every index, with the same element $g$ after appropriate conjugators. Conjugacy is explicitly over ${\mathbb{C}} G$; it is not silently changed to ${\mathbb{Q}} G$ or integral conjugacy.

Short English statement

The three questions concern recovering a finite group from its group algebra under an unspecified complement condition, nonvanishing of a particular partial augmentation for general linear groups, and whether a specially constructed mutually conjugate family of units is conjugate to group elements. Their missing maps and construction hypotheses prevent a complete unified theorem statement.

Sources

• Local UnsolvedMath registry, permanent ID $30000881$, 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 "rational conjugacy for nonsolvable groups": https://arxiv.org/abs/1305.7419.
• Supplied reference labelled "Zassenhaus counterexamples": https://doi.org/10.1016/j.aim.2018.10.004.
• The supplied corrected_verified note says all three questions were retained. The assessment checked 2026-08-21 explicitly says the record conflates modular algebra-isomorphism, partial-augmentation, and rational-conjugacy problems, reports substantial partial results, and labels the record partially solved. This is dated supplied triage; the counterexample reference and precise scope of remaining general questions require separate verification.