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The modular isomorphism problem: a survey. (English) Zbl 1535.20021

This survey is recommended to everyone curious to any aspect of the modular isomorphism problem. The article is written in an accessible way and surveys a plethora of facets of the group algebra of a \(p\)-group \(P\) over a field \(F\) of characteristic \(p\), called modular group algebra. The focus of the survey is on: “which invariants are determined by the \(F\)-algebra structure of the group algebra \(FP\)?”.
The invariants surveyed are not limited to group theoretical ones. For example also connections and implications to the associated Lie algebra of Lazard (§4, §6), block theory (§2.2), group cohomology (§9.1), the Lie structure of the first Hochschild cohomology (§9.2), normal complements (§6.2, §9.6) and some profinite aspects (§9.4) are discussed. Furthermore, methods and concepts are often illustrated with examples.
The article starts with a survey on the various isomorphism problems for group rings formulated in the literature. Such problem, denoted (\(R\)-IP), asks whether a finite group \(G\) is determined by its group ring over \(R\) for \(R\) some specific choice of commutative ring. In case of \(FP\) one speaks of the Modular Isomorphism Problem (MIP). In this part, also some of the main contributions of the last \(80\) years are mentioned. Most interestingly, also for an expert in group rings, in Section 2 the author emphasizes the contrasting behaviour of the modular and integral group ring, culminating for the reader to understand the intriguing quote of Sandling: “the perspectives to study (\(\mathbb{Z}\)-IP) is notable by their absence in the situation of (MIP)”.
Next, after a compact but rather complete overview of the state-of-the-art on MISO in Section 3, the article gives a glimpse into the methods. First, in Section 4, dimension subgroups and the Bruaer-Jennings-Zassenhaus series is recalled. In particular, the reader will learn about the associated Lie algebra of Lazard and the Jennings basis. As already becomes apparent from Section 5 and 6, the later concepts have been instrumental in many advances on (MIP).
In Section 7 an explicit construction and proof is presented of the recent counter-example to (MIP) [D. García-Lucas et al., J. Reine Angew. Math. 783, 269–274 (2022; Zbl 1514.20019)]. The final paragraphs of the section points out in how many surprising ways this counter-example is minimal and thus subtle. Comparing the groups constructed, one may filter non-invariants of the modular group algebra. This is done in Section 8 where a survey on non-invariants is included. Hereby, the reviewer would like to advertise two remaining open problems: (i) (MISO) for odd groups and (ii) the question whether \(p\)-groups are determined by their group algebra over all fields. As surveyed, those problems will require new perspectives. Section 9 luckily ends with various interesting other perspectives.
In sum, the survey is a very helpful service to the community and also deserves respect for its historical completeness.

MSC:

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20D15 Finite nilpotent groups, \(p\)-groups
16S34 Group rings
16U60 Units, groups of units (associative rings and algebras)

Citations:

Zbl 1514.20019
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References:

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