## Abelian invariants and a reduction theorem for the modular isomorphism problem.(English)Zbl 1533.16037

Let $$p$$ be a prime number, let $$G$$ be a finite $$p$$-group and let $$k$$ be a field of characteristic $$p$$. We say that $$G$$ satisfies the modular isomorphism problem if and only if for each group $$H$$, the group algebras $$kG$$ and $$kH$$ are isomorphic if and only if the groups $$G$$ and $$H$$ are isomorphic. The classic statement of the modular isomorphism problem asks whether this property holds for every finite $$p$$-group $$G$$, under the additional hypothesis that $$k$$ is the field with $$p$$ elements. It already appeared in [R. Brauer, Lect. Modern Math. 1, 133–175 (1963; Zbl 0124.26504)], and the first partial positive result goes back to [W. E. Deskins, Duke Math. J. 23, 35–40 (1956; Zbl 0075.23905)]. Even though the statement of the modular isomorphism problem is now known to be false in general due to the example provided in [D. García-Lucas et al., J. Reine Angew. Math. 783, 269–274 (2022; Zbl 1514.20019)], for odd primes it remains open.
In the paper under review the authors provide the first known “reduction theorem” for the modular isomorphism problem. Here by reduction theorem we mean a result that allows to answer the mentioned problem over the class of all finite $$p$$-groups studying only a proper subclass. For them (see Theorem 4.1), this subclass is the one of finite $$p$$-groups without elementary abelian direct factors. A simplified version of this reduction is stated by the authors in Theorem A, which reads as follows: if $$G, H$$ and $$E$$ are finite $$p$$-groups and moreover $$E$$ is elementary abelian, then $$k(G\times E)\cong k(H\times E)$$ implies that $$kG\cong kH$$.
A group theoretical feature of $$G$$ is said to be determined by its group algebra over $$k$$ if for any group $$H$$ such that $$kG \cong kH$$, the group $$H$$ has the same feature. The authors show in Theorem B that the isomorphism types of the following abelian subquotients of $$G$$ are determined by the group algebra $$kG$$:
$$G/\gamma(G)\Omega_n(\mathrm{Z}(G))$$,
$$\gamma(G)\Omega_n(\mathrm{Z}(G))/\gamma(G)$$,
$$\mathrm{Z}(G)\cap \mho_n(G) \gamma(G)$$,
$$\mathrm{Z}(G)/\mathrm{Z}(G)\cap \mho_n(G)\gamma(G)$$,
where $$\gamma(G)$$ denotes the derived subgroup of $$G$$, $$\mathrm{Z}(G)$$ denotes the center of $$G$$, and for a subset $$X$$ of $$G$$, one denotes $$\Omega_n(X)=\left\langle x\in X: x^{p^n}=1 \right\rangle$$ and $$\mho_n(X)=\left\langle x^{p^n} : x\in X \right \rangle$$. As an application of these new invariants, the authors show that modular isomorphism problem has positive answer for a $$2$$-group $$G$$ provided that $$\mathrm{Z}(G)$$ is cyclic and $$G/\mathrm{Z}(G)$$ is dihedral.
It should be mentioned that inspired by this paper the reviewer has generalized Theorem A in [D. García-Lucas, Mediterr. J. Math. 21, No. 1, Paper No. 18, 21 p. (2024; Zbl 1531.20026)] by dropping the “elementary” hypothesis when the field $$k$$ has $$p$$ elements, i.e., reducing the modular isomorphism problem to finite $$p$$-groups without abelian direct factors.

### MSC:

 16S34 Group rings 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 20D15 Finite nilpotent groups, $$p$$-groups

### Citations:

Zbl 0124.26504; Zbl 0075.23905; Zbl 1514.20019; Zbl 1531.20026

SmallGrp; GAP
Full Text:

### References:

 [1] Bagiński, C., Modular group algebras of 2-groups of maximal class, Commun. Algebra, 20, 1229-1241 (1992) · Zbl 0751.20004 [2] Bagiński, C.; Konovalov, A., The Modular Isomorphism Problem for Finite p-Groups with a Cyclic Subgroup of Index $$p^2$$, Groups St. Andrews 2005, vol. 1, 186-193 (2007), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1117.20003 [3] Bagiński, C.; Kurdics, J., The modular group algebras of p-groups of maximal class II, Commun. Algebra, 47, 761-771 (2019) · Zbl 1472.20003 [4] Besche, H. U.; Eick, B.; O’Brien, E.; GAP Team, T., SmallGrp – the GAP small groups library, version 1.4.2 (2020) [5] Broche, O.; del Río, Á., The modular isomorphism problem for two generated groups of class two, Indian J. Pure Appl. Math., 52, 721-728 (2021) · Zbl 1495.16035 [6] Carlson, J. F., Periodic modules over modular group algebras, J. Lond. Math. Soc. (2), 15, 431-436 (1977) · Zbl 0365.20015 [7] Carlson, J. F.; Townsley, L.; Valeri-Elizondo, L.; Zhang, M., Cohomology Rings of Finite Groups (2003), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 1056.20039 [8] Eick, B.; Konovalov, A., The Modular Isomorphism Problem for the Groups of Order 512, Groups St Andrews 2009 in Bath, vol. 2, 375-383 (2011), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1231.20002 [9] Evens, L., The Cohomology of Groups (1991), Clarendon Press: Clarendon Press Oxford · Zbl 0742.20050 [10] GAP - groups, algorithms, and programming, version 4.11.1 (2021) [11] García-Lucas, D., The modular isomorphism problem and abelian direct factors [12] García-Lucas, D.; Margolis, L.; del Río, Á., Non-isomorphic 2-groups with isomorphic modular group algebras, J. Reine Angew. Math., 783, 269-274 (2022) · Zbl 1514.20019 [13] Green, D. J.; King, S. A., The cohomology of finite p-groups (2015) [14] Hall, M.; Senior, J. K., The Groups of Order $$2^n(n \leq 6) (1964)$$, The Macmillan Company: The Macmillan Company New York · Zbl 0192.11701 [15] Hertweck, M.; Soriano, M., On the modular isomorphism problem: groups of order 2^6, (Groups, Rings and Algebras (2006), American Mathematical Society: American Mathematical Society Providence, RI), 177-213 · Zbl 1120.20005 [16] Huppert, B., Endliche Gruppen I (1967), Springer-Verlag: Springer-Verlag Berlin · Zbl 0217.07201 [17] Külshammer, B., Bemerkungen über die Gruppenalgebra als symmetrische Algebra. II, J. Algebra, 75, 59-69 (1982) · Zbl 0488.16010 [18] Macdonald, I. G., Symmetric Functions and Hall Polynomials (1995), Oxford University Press: Oxford University Press New York · Zbl 0899.05068 [19] Margolis, L.; Moede, T., ModIsomExt – an extension of ModIsom, version 1.0.0 (2020) [20] Margolis, L.; Stanojkovski, M., On the modular isomorphism problem for groups of class 3 and obelisks, J. Group Theory, 25, 163-206 (2022) · Zbl 1490.20006 [21] Navarro, G.; Sambale, B., On the blockwise modular isomorphism problem, Manuscr. Math., 157, 263-278 (2018) · Zbl 1499.20002 [22] Passman, D. S., The Algebraic Structure of Group Rings (1977), Wiley: Wiley New York · Zbl 0368.16003 [23] Ruiz, A.; Viruel, A., Cohomological uniqueness, Massey products and the modular isomorphism problem for 2-groups of maximal nilpotency class, Trans. Am. Math. Soc., 365, 3729-3751 (2013) · Zbl 1278.55026 [24] Sakurai, T., The isomorphism problem for group algebras: a criterion, J. Group Theory, 23, 435-445 (2020) · Zbl 1471.20001 [25] Sandling, R., The isomorphism problem for group rings: a survey, (Orders and Their Applications. Orders and Their Applications, Oberwolfach, 1984 (1985), Springer: Springer Berlin), 256-288 · Zbl 0565.20005 [26] Sandling, R., The modular group algebra problem for metacyclic p-groups, Proc. Am. Math. Soc., 124, 1347-1350 (1996) · Zbl 0844.20003 [27] Sehgal, S. K., Topics in Group Rings (1978), Marcel Dekker: Marcel Dekker New York · Zbl 0411.16004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.