Computing automorphism groups and testing isomorphisms for modular group algebras.
type: Item
instance of: scholarly article
title: Computing automorphism groups and testing isomorphisms for modular group algebras.
zbMATH Open document ID: 1163.20005
author: B. Eick
published in: Journal of Algebra
publication date: 2008-12-12T00:00:00Z
review text: The modular isomorphism problem of group algebras of finite \(p\)-groups over the prime field is still open. The aim of the paper is to provide a new algorithm for checking the existence of an isomorphism between the group algebras. There is already such an algorithm by \textit{M. Wursthorn}, [J. Symb. Comput. 15, No. 2, 211-227 (1993; Zbl 0782.20001)]. The new algorithm is effective and applicable for any nilponent finite dimensional algebra over a finite field. The elements of the algebra are identified with vectors of the coordinates, and multiplication is determined by the structure constant table. Two algebras are isomorphic if there is an invertible \(d\times d\) matrix compatible with multiplication. An isomorphism invariant table of canonical form is determined by induction according to the powers of the radical. The method uses the so-called covering algebra, analogous to the \(p\)-covering group of \(p\)-group algorithms due to \textit{M. F. Newman}, [Group Theory, Proc. Miniconf. Canberra 1975, Lect. Notes Math. 573, 73-84 (1977; Zbl 0519.20018)]. The input of the algorithm is the table of structure constants, the output is the canonical form, an isomorphism between these tables, and the generators and the order of the automorphism group. The algorithm was implemented in the computer algebra system GAP and was applied to check the isomorphism problem for groups of order \(2^8\) (more than 50000 in number) and \(3^6\) without yielding a counterexample. Previously the cases \(2^7\) and \(p^5\) had been checked.
reviewed by: János Kurdics
Mathematics Subject Classification ID: 20C40
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16N40
zbMATH DE Number: 5382249
zbMATH Keywords: modular isomorphism problem
zbMATH Keywords: automorphism groups
zbMATH Keywords: isomorphism testing
zbMATH Keywords: nilpotent algebras
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: algorithms
zbMATH Keywords: groups of order \(p^6\)
zbMATH Keywords: symbolic computations
zbMATH Keywords: modular group rings
describes a project that uses: Sophus
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.05.002
OpenAlex ID: W2073101508
cites work: A MILLENNIUM PROJECT: CONSTRUCTING SMALL GROUPS
cites work: Q4247795
cites work: Deux groupes finis distincts ayant la même algèbre de groupe sur tout corps
cites work: Finite Abelian groups with isomorphic group algebras
cites work: CONSTRUCTING AUTOMORPHISM GROUPS OF p-GROUPS
cites work: On \(p\)-groups forming Brauer pairs.
cites work: Q3869534
cites work: A counterexample to the isomorphism problem for integral group rings
cites work: Parametrization of central Frattini extensions and isomorphisms of small group rings.
cites work: Q4650358
cites work: Q4312071
cites work: Q3669627
cites work: Descriptions of groups of prime-power order
cites work: APPLICATION OF COMPUTERS TO QUESTIONS LIKE THOSE OF BURNSIDE, II
cites work: Isomorphism testing for \(p\)-groups
cites work: Q4843626
cites work: Isomorphism of modular group algebras
cites work: Q3961610
cites work: Isomorphisms of p-adic group rings
cites work: Automorphisms and nonabelian cohomology: An algorithm
cites work: Q4342753
cites work: The modular group algebra problem for metacyclic 𝑝-groups
cites work: A Computer-Based Approach to the Classification of Nilpotent Lie Algebras
cites work: Q3679379
cites work: Isomorphisms of modular group algebras: An algorithm and its application to groups of order \(2^ 6\)
DOI: 10.1016/J.JALGEBRA.2008.05.002
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description: scientific article; zbMATH DE number 5382249