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We analyse the complexity of constructing involution centralisers in unitarygroups over fields of odd order. In particular, we prove logarithmic bounds onthe number of random elements required to generate a subgroup of thecentraliser of a strong involution that contains the last term of its derivedseries. We use this to strengthen previous bounds on the complexity ofrecognition algorithms for unitary groups in odd characteristic. Our approachgeneralises and extends two previous papers by the second author andcollaborators on strong involutions and regular semisimple elements of lineargroups.

Involution centralisers in finite unitary groups of odd characteristic