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

Fixed point spaces in actions of classical algebraic groups