Special functions in Clifford analysis and axial symmetry
In our previous paper [Rend. Circ. Mat. Palermo, II Ser. Suppl. 6, 259- 269 (1984; Zbl 0564.30036)], we proved a general Laurent expansion for monogenic functions in symmetric domains of \({\mathbb{R}}^{m+1}\), depending on the kind of symmetry involved. In this paper we consider axial symmetric domains and we apply our results in order to introduce new speciat \({\mathcal H}\) be a principal series of G. The Gaschütz-module A of G is defined as the direct product of all splitting p-chief factors of \({\mathcal H}\). A is in a natural manner a G-(right)-module. There exists a ``canonical'' crossed homomorphism \(\psi\) of G onto A; ker \(\psi\) \(=D\) is a p-prefrattini-group of G (i.e. the intersection of a set of complements, one for each complemented p-chief-factor of \({\mathcal H})\). The group ring \({\mathbb{F}}_ p[A]\) is made a G-module by \(a\circ g:=a^ g\psi (g)\), \(a\in A\), \(g\in G\) where \(a^ g\) indicates the natural operation of G on A. This module is called \({\mathbb{F}}_ p[A]_{\psi}.\) Then the following holds: a) There is a G epimorphism \({\hat \psi}\): \(P_ 1\to {\mathbb{F}}_ p[A]_{\psi}\), ker \({\hat \psi}\cong P_ 1(D)J_ D\otimes_ D{\mathbb{F}}_ p[G]\), where \(J_ D\) is the radical of \({\mathbb{F}}_ p[D]\). b) \({\hat \psi}\) maps \(J_ G\) onto \(J_{A,\psi}\) the augmentation ideal of \({\mathbb{F}}_ p[A]_{\psi}\). c) The mapping \(\pi\) : \(J_{a,\psi}\to A\), given by a-1\(\to^{\pi}a\) is a G-epimorphism. d) \(\tau =\pi\). \({\hat \psi}\) maps \(P_ 1J_ G\) onto A and ker \(\tau\) \(=P_ 1J^ 2_ G.\) Gaschütz's theorem: \(P_ 1J_ G/P_ 1J^ 2_ G\cong A\) is a consequence. Moreover, because of the fact that the composition factors of \({\mathbb{F}}_ p[A]_{\psi}\) coincide with the composition factors of the natural G-module \({\mathbb{F}}_ p[A]\) all these are composition factors of \(P_ 1\). As a further application one gets a simple proof of the theorem of Green and Hill. More information on \(P_ 1(G)\) is given.
scientific article; zbMATH DE number 4032058
scientific article