A conjecture of Abe-Iiyori and finite orthogonal simple groups with Witt index \(n\) \(P\Omega^+_{2n}(q)\). III. (Q874796)

[For part II cf. \textit{M.-C. Xu}, Sci. China, Ser. A 49, No. 9, 1158-1164 (2006; Zbl 1116.20008).] For a finite group \(G\) let \(\text{Ord}(S_{\text{Sol}}(G))\) denote the set of orders of solvable subgroups of \(G\). A conjecture due to S. Abe and N. Iiyori states that if \(S\) is a finite simple group and \(G\) is a finite group, then \(\text{Ord}(S_{\text{Sol}}(G))=\text{Ord}(S_{\text{Sol}}(S))\) if and only if \(G\cong S\). In the present paper, the authors prove the conjecture in the case of \(S=P\Omega _{2n}^+(q)\), where \(P\Omega _{2n}^+(q)\) is the finite orthogonal group of Witt index \(n\).