Parametrization of central Frattini extensions and isomorphisms of small group rings.
type: Item
instance of: scholarly article
title: Parametrization of central Frattini extensions and isomorphisms of small group rings.
zbMATH Open document ID: 1120.20004
author: Martin Hertweck
author: Marcos Soriano
published in: Israel Journal of Mathematics
publication date: 2007-04-23T00:00:00Z
review text: By definition, the small group ring of a group extension (*) \(V\hookrightarrow E\twoheadrightarrow H\), with \(V\) central \(p\)-elementary Abelian contained in the Frattini subgroup of \(E\) and with \(H\) a (fixed) \(p\)-group, is the quotient \(s(E,V)=\mathbb{F}_pE/\Delta(V)\Delta(E)\), where \(\Delta\) abbreviates the augmentation ideal (so \(\Delta(V)=\ker(\mathbb{F}_pV@>{\text{aug}}>>\mathbb{F}_p)\) and \(V\simeq\Delta(V)\mathbb{F}_pE/\Delta(V)\Delta(E)\)). Let \(\mathbb{F}(H)\) denote the class of all group extensions (*) as above. Each such extension determines an exact sequence (**) \(V\rightarrowtail s(V,E)\twoheadrightarrow\mathbb{F}_pE/V\). This paper describes the equivalence classes of the extensions (**) (or, equivalently, the isomorphism classes of the corresponding small group rings) by means of the action of \(\text{Out}(\mathbb{F}_pH)\) on the kernel \(V\) of a certain universal extension (which arises from an appropriate free presentation of \(H\)). Restricting the action to \(\text{Out}(H)\) then yields precisely the isomorphism classes of the extensions (*) in \(\mathbb{F}(H)\). As a corollary, the modular isomorphism problem for small group rings is solved. The paper may thus be viewed as an advertisement for an inductive approach to the modular isomorphism problem. Many examples accompany the discussion of the relation between (*) and (**). Related literature is e.g.~[\textit{I. Borge, O. Laudal}: Preprint series in Pure Mathematics 19 (2002), 25 (2003); \textit{M. Hertweck, M. Soriano}: ``Central \(p\)-group extensions: Laudal obstruction spaces revisited'' (2005), preprint; \textit{F. Röhl}, Can. J. Math.~42, No. 3, 383-394 (1990; Zbl 0711.20005)].
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 5145471
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: small group rings
zbMATH Keywords: modular group algebras
zbMATH Keywords: modular isomorphism problem
zbMATH Keywords: group extensions
zbMATH Keywords: augmentation ideals
reviewed by: Jürgen Ritter
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: s11856-006-0003-5
OpenAlex ID: W2028440407
cites work: The word of others
cites work: Periodic Modules over Modular Group Algebras
cites work: On the Modular Group Ring of a p-Group
cites work: The Annihilator of Radical Powers in the Modular Group Ring of A p-Group
cites work: Q5648522
cites work: Endliche Gruppen I
cites work: Q3328735
cites work: The Structure of the Group Ring of a p-Group Over a Modular Field
cites work: Isomorphism of modular group algebras
cites work: Q4145843
cites work: Isomorphism of group rings
cites work: Isomorphisms of p-adic group rings
cites work: On the isomorphism problem for group rings and completed augmentation ideals
cites work: On Automorphisms of Complete Algebras And The Isomorphism Problem for Modular Group Rings
cites work: Ein Verfahren, jeder endlichen \(p\)-Gruppe einen Lie-Ring mit der Charakteristik \(p\) zuzuordnen
DOI: 10.1007/S11856-006-0003-5
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description: scientific article; zbMATH DE number 5145471