Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 46, No. 1, pp. 169177 (2005) 

Using the Frattini subgroup and independent generating sets to study RWPri geometriesClaude Archer, Philippe Cara and Jan KrempaUniversité Libre de Bruxelles, C.P.165/11 Physique et Mathématique Faculté des Sciences Appliquées, avenue F.D. Roosevelt 50, 1050 Bruxelles, Belgium, email: carcher@ulb.ac.be; Vrije Universiteit Brussel, Department of Mathematics, Pleinlaan 2, B1050 Brussel, Belgium, email: pcara@vub.ac.be; Warsaw University, Institute of Mathematics, Banacha 2, 02097 Warszawa, Poland, email: jkrempa@mimuw.edu.plAbstract: In [CC] Cameron and Cara showed a relationship between independent generating sets of a group $G$ and RWPri geometries for $G$. We first notice a connection between such independent generating sets in $G$ and those in the quotient $G/\Phi (G)$, where $\Phi (G)$ is the Frattini subgroup of $G$. This suggests a similar connection for RWPri geometries. We prove that there is a onetoone correspondence between the $\rwp$ geometries of $G$ and those of $G/\Phi (G)$. Hence only RWPri geometries for Frattini free groups have to be considered. We use this result to show that RWPri geometries for $p$groups are direct sums of rank one geometries. We also give a new test which can be used when one wants to enumerate RWPri geometries by computer. [CC] Cameron, P. J.; Cara, Ph.: Independent generating sets and geometries for symmetric groups. J. Algebra {\bf 258} (2002), 641650. Full text of the article:
Electronic version published on: 11 Mar 2005. This page was last modified: 4 May 2006.
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