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It is known that a finite non-abelian group G has a proper centralizer of order 
if, for example, |G| is even and |Z(G)| is odd, or whenever G is solvable. Often the exponent 
can be improved to 
, for example when G is supersolvable, or metabelian, or |G = pαqβ. Here we show more generally that this improvement is possible in many situations where G is factorizable into the product of two subgroups. In particular, much more evidence is presented to support the conjecture that some proper centralizer has order 
whenever G is a finite non-abelian solvable group.
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