Towards Thompson's conjecture for alternating and symmetric groups. (Q254326)

For a finite group \(G\), let \(N(G)\) denote the set of conjugacy class sizes of \(G\). In 1987 Thompson posed the following conjecture concerning \(N(G)\) (see also Question 12.38 in ``Kourovka notebook'' [1992; Zbl 0831.20003]): If \(L\) is a finite simple nonabelian group, \(G\) is a finite group with trivial center, and \(N(G)=N(L)\), then \(G\cong L\). Thompson's conjecture was proved valid for many finite simple groups of Lie type (see the papers of \textit{N. Ahanjideh} [J. Algebra 344, No. 1, 205-228 (2011; Zbl 1247.20015); Algebra Logic 51, No. 6, 451-478 (2013; Zbl 1309.20019); translation from Algebra Logika 51, No. 6, 683-721 (2012)] and \textit{N. Ahanjideh} and \textit{M. Ahanjideh} [Commun. Algebra 41, No. 11, 4116-4145 (2013; Zbl 1285.20010)]) and for all sporadic finite simple groups (see the paper of \textit{G. Chen} [J. Algebra 185, No. 1, 184-193 (1996; Zbl 0861.20018)]).NEWLINENEWLINE Denote the alternating group of degree \(n\) by \(\text{Alt}_n\) and the symmetric group of degree \(n\) by \(\text{Sym}_n\). Thompson's conjecture has been confirmed by \textit{S. H. Alavi} and \textit{A. Daneshkhah} for the groups \(\text{Alt}_n\) where \(n\in\{p,p+1,p+2\}\) for prime \(p\geq 5\) [J. Appl. Math. Comput. 17, No. 1-2, 245-258 (2005; Zbl 1066.20012)], by the author for \(\text{Alt}_{10}\) [Algebra Logic 51, No. 2, 111-127 (2012; Zbl 1270.20010); translation from Algebra Logika 51, No. 2, 168-192 (2012)], by \textit{A. V. Vasil'ev} for \(\text{Alt}_{16}\) [Sib. Èlektron. Mat. Izv. 6, 457-464 (2009; Zbl 1289.20057)] and by \textit{M. Xu} for \(\text{Alt}_{22}\) [Front. Math. China 8, No. 5, 1227-1236 (2013; Zbl 1281.20018)].NEWLINENEWLINE In the given paper, the author shows that if \(G\) is a finite group such that \(N(G)=N(\text{Alt}_n)\) for \(n>4\), or \(N(G)=N(\text{Sym}_n)\) for \(n>22\), then \(G\) is non-solvable.