Let G be a simple classical algebraic group over an algebraically closed field K of characteristic p ? 0, and let H be a maximal closed non-subspace subgroup of G. Given such a pair (G, H), we obtain a close to best possible upper bound for the ratio dim(xG ? H) / =dim xG, where x ? G is a semisimple or unipotent element of prime order. We apply this result to the study of fixed point spaces
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