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Let t be an involution in GL(n,q) whose fixed point space E+ has dimension k between n/3 and 2n/3. For each g∈GL(n,q) such that ttg has even order, 〈ttg〉 contains a unique involution z(g) which commutes with t. We prove that, with probability at least c/logn (for some c>0), the restriction z(g)|E+ is an involution on E+ with fixed point space of dimension between k/3 and 2k/3. This result has...
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