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NIELSEN EQUIVALENCE OF GENERATING PAIRS OF SL(2,q)

Published online by Cambridge University Press:  25 February 2013

DARRYL MCCULLOUGH
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA e-mail: dmccullough@math.ou.edu
MARCUS WANDERLEY
Affiliation:
Departmento de Matematica, Universidade Federal de Pernambuco, Av. Prof. Luiz Freire, s/n Cid. Universitaria-Recife-PE, CEP 50.740-540, Brazil e-mail: marcus@dmat.ufpe.br
Corresponding
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Abstract

We present several conjectures which would describe the Nielsen equivalence classes of generating pairs for the groups SL(2,q) and PSL(2,q). The Higman invariant, which is the union of the conjugacy classes of the commutator of a generating pair and its inverse, and the trace of the commutator play key roles. Combining known results with additional work, we clarify the relationships between the conjectures, and obtain various partial results concerning them. Motivated by the work of Macbeath (A. M. Macbeath, Generators of the linear fractional groups, in Number theory (Proc. Sympos. Pure Math., vol. XII, Houston, TX, 1967) (American Mathematical Society, Providence, RI, 1969), 14–32), we use another invariant defined using traces to develop algorithms that enable us to verify the conjectures computationally for all q up to 101, and to prove the conjectures for a highly restricted but possibly infinite set of q.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

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