Using the Frattini Subgroup and Independent Generating Sets to Study RWPRI Geometries

In [4], 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/Φ(G), where Φ(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 RWPri geometries of G and those of G/Φ(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. Corresponding author; postdoctoral fellow of the Fund for Scientific Research-Flanders (Belgium) (F.W.O.-Vlaanderen). 0138-4821/93 $ 2.50 c © 2005 Heldermann Verlag 170 C. Archer, Ph. Cara, J. Krempa: Using the Frattini Subgroup and . . .