Beiträge zur Algebra und GeometrieContributions to Algebra and Geometry
Vol. 46, No. 1, pp. 169-177 (2005)
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Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 46, No. 1, pp. 169-177 (2005) |
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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, e-mail: carcher@ulb.ac.be; Vrije Universiteit Brussel, Department of Mathematics, Pleinlaan 2, B-1050 Brussel, Belgium, e-mail: pcara@vub.ac.be; Warsaw University, Institute of Mathematics, Banacha 2, 02-097 Warszawa, Poland, e-mail: 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 one-to-one 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), 641--650. Full text of the article:
Electronic version published on: 11 Mar 2005. This page was last modified: 4 May 2006.
© 2005 Heldermann Verlag
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