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Is it true that in a simple group with abelian Sylow 2-subgroups a Sylow 2-subgroup is not centralized by a nontrivial element of odd order? Maybe the paper of John Walter "The characterization of finite groups with abelian Sylow 2-subgroups" is useful.

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Yes. According to the classification by Walter, the nonabelian finite simple groups with abelian Sylow $2$-subgroups are:

(i) ${\rm PSL}(2,q)$ with $q \equiv 3,5 \bmod 8$;

(ii) ${\rm PSL}(2,2^n)$ with $n \ge 2$;

(iii) The Ree groups $R(3^e) = {}^2G_2(3^e)$ with $e$ odd;

(iv) The sporadic Janko group $J_1$.

The Sylow $2$-subgroups are elementary abelian in all four cases, with orders $2^2$, $2^n$, $2^3$ and $2^3$, respectively.

Their normalizers have orders $12$ ($\cong A_4$), $2^n(2^n-1)$, $24$ ($\cong C_2 \times A_4$), and $168$, respectively, and they are equal to their own centralizers in all four cases, so they are centralized by no nontrivial element of odd order.

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