Recognising the Suzuki groups in their natural representations

Abstract

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 2 m + 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 ) log log ( q ) ) field operations. The latter step needs an oracle for the discrete logarithm problem in F q . We also produce a recognition algorithm for Sz ( q ) = 〈 X 〉 . This is a Las Vegas algorithm with running time O{} ( | X | 2 ) field operations. Finally, we give a Las Vegas algorithm that, given 〈 X 〉 h = Sz ( q ) for some h ∈ GL ( 4 , q ) , finds some g such that 〈 X 〉 g = Sz ( q ) . The running time is O{} ( log ( q ) log log ( q ) + | X | ) field operations. Implementations of the algorithms are available for the computer system Magma.