Fixed points of coprime operator groups

Let m be a positive integer and A an elementary abelian group of order q^r with r greater than or equal to 2 acting on a finite q'-group G. We show that if for some integer d such that 2^{d} is less than or equal to (r-1) the dth derived group of C_{G}(a) has exponent dividing m for any nontrivial element a in A, then $G^{(d)}$ has {m,q,r}-bounded exponent and if $γ_{r-1}(C_G(a))$ has exponent dividing m for any nontrivial element a in A, then $γ_{r-1}(G)$ has {m,q,r}-bounded exponent.

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