A criterion for the failure of local balance of some simple groups of Lie type

A finite simple nonabelian group $K$ is called locally balanced (locally 1-balanced) with respect to a prime$p$ if $O_{p'}(C_G(x))=1$ for any element$x$ of order$p$ from $G\simeq\rm Aut\,(K)$. Finite simple nonabelian groups that are not locally balanced were described in the famous Theorem 7.7.1 from The Classification of the Finite Simple Groups by Gorenstein, Lyons, and Solomon. However, there is a gap in statement(d) of that theorem, which is also present in the proof. In this connection, we prove the following theorem. Theorem. Suppose that $G$ is a finite almost simple group, $K=\rm Soc\,(G)$ is a group of Lie type over a field of characteristic$r$, and $x$ is an element of a prime order $p\ne r$ from $G$ that induces on $K$ a non-inner-diagonal automorphism. Then the following conditions are equivalent: $(1)$ $O_{p'}(C_G(x))\ne 1;$ $(2)$ $x$ induces a field automorphism on $K$ and $(|C_K(x)|,p)=1$. The theorem gives a criterion for the local 1-imbalance of groups of Lie type from statement(d) of the mentioned Theorem 7.7.1 with a non-inner-diagonal automorphism. The criterion can be used to construct a countable series of counterexamples to this statement for any simple nonabelian group of Lie type.