Non-isomorphic 2-groups with isomorphic modular group algebras

Diego, García-Lucas,; Margolis, L.; Río, Ángel del

doi:10.1515/crelle-2021-0074

Cited by 14 publications

(6 citation statements)

References 40 publications

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“…we get a final answer on (MIP) for a given pair of groups G and H and not just restrictions on possible properties of group bases. Indeed, proceedings along these lines the counterexamples in [GLMdR22] were discovered.…”

Section: Embedding the Group Basis In Quotientsmentioning

confidence: 98%

“…We describe the counterexamples to (MIP) which have been recently obtained [GLMdR22]. As the proof is short and easy to understand, we include it in full.…”

Section: The Counterexamplesmentioning

confidence: 99%

“…But in the last years it was again attacked independently by several researchers [BK19, Sak20, MM20b, MS22, BdR21, MSS21] and this culminated in the solution of the problem in its general form: Theorem 1.1. [GLMdR22] There are non-isomorphic finite 2-groups G and H such that kG ∼ = kH for any field k of characteristic 2.…”

Section: Introductionmentioning

confidence: 99%

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The Modular Isomorphism Problem: a survey

Margolis

2022

Preprint

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The Modular Isomorphism Problem asks if an isomorphism of group algebras of two finite p-groups G and H over a field of characteristic p, implies an isomorhism of the groups G and H. We survey the history of the problem, explain strategies which were developed to study it and present the recent negative solution of the problem. The problem is also compared to other isomorphism problems for group rings and various question remaining open are included.

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Section: Embedding the Group Basis In Quotientsmentioning

confidence: 98%

“…We describe the counterexamples to (MIP) which have been recently obtained [GLMdR22]. As the proof is short and easy to understand, we include it in full.…”

Section: The Counterexamplesmentioning

confidence: 99%

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The Modular Isomorphism Problem: a survey

Margolis

2022

Preprint

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“…0, 0, 0, 0, 1, 1). Note that, for p > 2, one always has σ 1 = σ 2 = 1 and therefore the counterexample from [16] does not have a direct equivalent in odd characteristic. We will see that Theorem B is actually equivalent to the following.…”

Section: Theorem B Let K Be a Field Of Odd Characteristic P And Let G...mentioning

confidence: 99%

“…[1-4, 6, 13-15, 19, 20, 22, 25, 27-30, 34, 36, 38, 40, 41, 45-47, 49]. The first negative solution to the Modular Isomorphism Problem was given recently in the form of a series of pairs of non isomorphic 2-groups G m,n and H m,n which are 2-generated and have cyclic commutator subgroup satisfying kG m,n ∼ = kH m,n for every n > m > 2 and every field k of characteristic 2 [16]. However, if p is odd then the Modular Isomorphism Problem is still open, even in the class of 2-generated groups with cyclic commutator subgroup.…”

Section: Introductionmentioning

confidence: 99%

On Group Invariants Determined by Modular Group Algebras: Even Versus Odd Characteristic

Diego

Río

Stanojkovski

2022

Algebr Represent Theor

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Let p be a an odd prime and let G be a finite p-group with cyclic commutator subgroup $G^{\prime }$ G ′ . We prove that the exponent and the abelianization of the centralizer of $G^{\prime }$ G ′ in G are determined by the group algebra of G over any field of characteristic p. If, additionally, G is 2-generated then almost all the numerical invariants determining G up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of $G^{\prime }$ G ′ is determined. These claims are known to be false for p = 2.

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The Modular Isomorphism Problem and Abelian Direct Factors

García-Lucas

2023

Mediterr. J. Math.

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Let p be a prime and let G be a finite p-group. We show that the isomorphism type of the maximal abelian direct factor of G, as well as the isomorphism type of the group algebra over $${{\mathbb {F}}}_p$$ F p of the non-abelian remaining direct factor, if existing, are determined by $${{\mathbb {F}}}_p G$$ F p G , generalizing the main result in Margolis et al. (Abelian invariants and a reduction theorem for the modular isomorphism problem, Journal of Algebra 636, 533-559 (2023)) over the prime field. To do this, we address the problem of finding characteristic subgroups of G such that their relative augmentation ideals depend only on the k-algebra structure of kG, where k is any field of characteristic p, and relate it to the modular isomorphism problem, extending and reproving some known results.

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Non-isomorphic 2-groups with isomorphic modular group algebras

Abstract: We provide non-isomorphic finite 2-groups which have isomorphic group algebras over any field of characteristic 2, thus settling the Modular Isomorphism Problem.

Cited by 14 publications

References 40 publications

The Modular Isomorphism Problem: a survey

The Modular Isomorphism Problem: a survey

On Group Invariants Determined by Modular Group Algebras: Even Versus Odd Characteristic

The Modular Isomorphism Problem and Abelian Direct Factors

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