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**On the modular isomorphism problem for groups of class \(3\) and obelisks.**
*(English)*
Zbl 1490.20006

Let \(G\) and \(H\) be finite \(p\)-groups and \(k\) be a finite field of characteristic \(p\). The modular isomorphism problem (MIP), which is still open, can be stated as follows: Does the isomorphism \(kG\cong kH\) of the modular group algebras of \(G\) and \(H\) imply that the groups \(G\) and \(H\) are isomorphic? Alternatively, if \(H\) is a subgroup of the unit group of \(kG\) such that \(kG\cong kH\), then \(H\) is said to be a group base of \(kG\). The MIP can then be reformulated as follows: are all group bases of a modular group algebra of a finite \(p\)-group isomorphic?

Since an explicit formulation of the MIP was given by Brauer in 1963, the problem has been studied by many researchers. Inspired by new points of view and approaches, the authors address the MIP for certain classes of groups from different directions and techniques. First, they employ the so-called small group algebra of \(kG\) to solve the MIP for two new classes of groups. Among the biggest merits of the use of this construction, two results stand out: Sandling’s solution of the MIP for groups with central elementary abelian derived subgroup, and likewise, M. A. M. Salim and R. Sandling’s solution for groups of order \(p^5\) [J. Aust. Math. Soc., Ser. A 61, No. 2, 229–237 (1996; Zbl 0874.20003)]. Even though the authors show, by means of an example, that the strategy of Salim and Sandling [loc. cit.] cannot be directly transferred to groups of order \(p^6\), the results in the paper under review cover, for \(p\) odd, \(5\) of \(43\) isoclinism classes of groups of order \(p^6\), while \(10\) other classes are covered by already known results. The authors themselves claim that the methods employed can be applied to other families of groups having maximal subgroups of relatively small class.

The paper continues with a new approach to derive properties of the lower central series of a finite \(p\)-group from the structure of the associated modular group algebra. Using original quotient-in-quotient embedding techniques, the authors show that, for certain \(2\)-generated groups of nilpotency class \(3\), the isomorphims types of the members of the lower central series of any group base are determined by \(kG\).

The paper concludes with the study of the MIP for a family of groups named \(p\)-obelisks. A \(p\)-obelisk is a finite non-abelian \(p\)-group \(P\) satisfying \(|P:\gamma_2(P)|=p^2\) and \(P^p=\gamma_3(P)\). These groups are hightlighted by recent computer-aided investigations for groups of order \(5^6\), and in some sense are among the groups of order \(5^6\) that are “closest” to being counterexamples to the MIP. The authors prove that if \(kG\cong KH\) and \(G\) is a \(p\)-obelisk, then so is \(H\).

Since an explicit formulation of the MIP was given by Brauer in 1963, the problem has been studied by many researchers. Inspired by new points of view and approaches, the authors address the MIP for certain classes of groups from different directions and techniques. First, they employ the so-called small group algebra of \(kG\) to solve the MIP for two new classes of groups. Among the biggest merits of the use of this construction, two results stand out: Sandling’s solution of the MIP for groups with central elementary abelian derived subgroup, and likewise, M. A. M. Salim and R. Sandling’s solution for groups of order \(p^5\) [J. Aust. Math. Soc., Ser. A 61, No. 2, 229–237 (1996; Zbl 0874.20003)]. Even though the authors show, by means of an example, that the strategy of Salim and Sandling [loc. cit.] cannot be directly transferred to groups of order \(p^6\), the results in the paper under review cover, for \(p\) odd, \(5\) of \(43\) isoclinism classes of groups of order \(p^6\), while \(10\) other classes are covered by already known results. The authors themselves claim that the methods employed can be applied to other families of groups having maximal subgroups of relatively small class.

The paper continues with a new approach to derive properties of the lower central series of a finite \(p\)-group from the structure of the associated modular group algebra. Using original quotient-in-quotient embedding techniques, the authors show that, for certain \(2\)-generated groups of nilpotency class \(3\), the isomorphims types of the members of the lower central series of any group base are determined by \(kG\).

The paper concludes with the study of the MIP for a family of groups named \(p\)-obelisks. A \(p\)-obelisk is a finite non-abelian \(p\)-group \(P\) satisfying \(|P:\gamma_2(P)|=p^2\) and \(P^p=\gamma_3(P)\). These groups are hightlighted by recent computer-aided investigations for groups of order \(5^6\), and in some sense are among the groups of order \(5^6\) that are “closest” to being counterexamples to the MIP. The authors prove that if \(kG\cong KH\) and \(G\) is a \(p\)-obelisk, then so is \(H\).

Reviewer: Antonio Beltrán Felip (Castellón)

### MSC:

20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |

16S34 | Group rings |

20D15 | Finite nilpotent groups, \(p\)-groups |

### Citations:

Zbl 0874.20003
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\textit{L. Margolis} and \textit{M. Stanojkovski}, J. Group Theory 25, No. 1, 163--206 (2022; Zbl 1490.20006)

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