Generation of finite classical groups by pairs of elements with large fixed point spaces

We study ‘good elements’ in finite 2n-dimensional classical groups G: namely t   is a ‘good element’ if o(t)o(t) is divisible by a primitive prime divisor of qn−1qn−1 for the relevant field order q, and t fixes pointwise an n  -space. The group SL2n(q)SL2n(q) contains such elements, and they are present in SU2n(q),Sp2n(q),SO2nε(q), only if n is odd, even, even, respectively. We prove that there is an absolute positive constant c such that two random conjugates of t generate G with probability at least c  , if G≠SO2nε(2) and G≠Sp2n(q)G≠Sp2n(q) with q   even. In the exceptional case G=Sp2n(q)G=Sp2n(q) with q even, two conjugates of t never generate G: in this case we prove that two random conjugates of t   generate a subgroup SO2nε(q) with probability at least c. The results underpin analysis of new constructive recognition algorithms for classical groups in even characteristic, which succeed where methods utilising involution centralisers are not available.