Finite groups whose Abelian subgroups have consecutive orders (Q1269426)

Let \(n\) be a positive integer. The finite group is called an \(OA_n\)-group (resp. an \(OC_n\)-group) if the possible orders of its Abelian (resp. cyclic) subgroups are \(\{1,2,\dots,n\}\), where it is assumed that all orders actually occur. \textit{R. Brandl} and \textit{W. Shi} [J. Algebra 143, No. 2, 388-400 (1991; Zbl 0745.20022)] have shown that \(OC_n\)-groups exist if and only if \(n\leq 8\). Using similar methods, the following is shown: Theorem. Let \(G\) be an \(OA_n\)-group. Then \(n\leq 6\) and \(G\) is isomorphic to \(S_k\) (\(k\leq 5\)) or the alternating groups \(A_4\), \(A_5\). In particular, there exist only finitely many \(OA_n\)-groups. The proof uses properties of the prime graph introduced by \textit{J. S. Williams} [J. Algebra 69, 487-513 (1981; Zbl 0471.20013)] and depends on the classification of all finite simple groups.