Recognising the Suzuki groups in their natural representations

Details Recognising the Suzuki groups in their natural representations Recognising the Suzuki groups in their natural representations

2006-08-09

arxiv.org/abs/math/0608210v1

J. Algebra 300 (1), 171-198, 2006

Mathematics

Group Theory

Scientific paper

10.1016/j.jalgebra.2006.02.010

Under the assumption of a certain conjecture, for which there exists strong experimental evidence, we produce an efficient algorithm for constructive membership testing in the Suzuki groups Sz(q), where q = 2^{2m + 1} for some m > 0, in their natural representations of degree 4. It is a Las Vegas algorithm with running time O{log(q)} field operations, and a preprocessing step with running time O{log(q) loglog(q)} field operations. The latter step needs an oracle for the discrete logarithm problem in GF(q). We also produce a recognition algorithm for Sz(q) = . This is a Las Vegas algorithm with running time O{|X|^2} field operations. Finally, we give a Las Vegas algorithm that, given ^h = Sz(q) for some h in GL(4, q), finds some g such that ^g = Sz(q). The running time is O{log(q) loglog(q) + |X|} field operations. Implementations of the algorithms are available for the computer system MAGMA.

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