Abstract
An involution in a finite n-dimensional classical group G over a field of odd order q is called (α, β)-balanced if the dimension of its fixed point subspace is between αn and βn. Balanced involutions play an important role in recent constructive recognition algorithms for finite classical groups in odd characteristic. For a given sequence 
of conjugacy classes of balanced involutions in G, a c-tuple (g
1, . . . , gc
) is a class-random sequence from 𝒳 if, for each i = 1, . . . , c, gi
is a uniformly distributed random element of 
, and the gi
are mutually independent. We show that there is a number c = c(α, β) such that for large enough n, for a given such sequence 𝒳 of length c, a class-random sequence from 𝒳 generates a subgroup containing the generalized Fitting subgroup of G with probability at least 1 – q
–n
.
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