Parametrization of Actions of the Automorphism Subgroup of a Cubic Tree on PL(F_p)

Abstract

The group is important because it is one of the seven finitely presented isomorphism types of subgroups of the full automorphism group (Aut(Γ₃)) of a cubic tree Γ₃. These seven groups act arc-transitively on the arcs of Γ₃ with a finite vertex stabilizer. In this article, we have parametrized the actions of on the projective line over the finite field, PL(F _p ), where p is the Pythagorean prime and has thus shown that there is only one coset diagram depicting the sole conjugacy class of these actions.

Key Words:

• Automorphism groups
• Conjugacy classes
• Connected graphs
• Coset diagrams
• Parametrizations
• Pythagorean primes
• Vertex stabilizers

2000 Mathematics Subject Classification:

• Primary 20G40
• Secondary 20B35, 05C25

ACKNOWLEDGMENT

Dedicated to the memory of Late Professor Graham Higman FRS.

Notes

Communicated by V. A. Artamonov.