Parametrization of central Frattini extensions and isomorphisms of small group rings.
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)].
scientific article; zbMATH DE number 5145471
scientific article