On the nilpotency of the solvable radical of a finite group isospectral to a simple group (Q2182110)

The spectrum of a finite group \(G\) is the set of the orders of its elements and two finite groups are said to be isospectral if their spectra coincide. It is known that \(L_3(3), U_3(3)\) and \(S_4(3)\) are the only finite nonabelian simple groups that have the same spectrum as some solvable group and that if a nonsolvable group \(G\) is isospectral to an arbitrary nonabelian simple, then the factor group \(G/K\), where \(K\) is the solvable radical of \(G,\) is an almost simple group. The authors are interested in the nilpotency of the solvable radical \(K\). They prove that if \(L\) is a finite nonabelian simple group distinct from the alternating group \(A_{10}\) and \(G\) is a finite nonsolvable group isospectral to \(L\), then the solvable radical \(K\) of \(G\) is nilpotent.