Huppert's conjecture for alternating groups (Q335613)

For a finite group \(G\) let \(\operatorname{cd}(G)\) denote the set of irreducible character degrees of \(G\). Huppert's conjecture states that if \(G\) and \(H\) are finite groups with \(G\) simple such that \(\operatorname{cd}(G)=\operatorname{cd}(H)\), then \(H\) is isomorphic to the direct product \(G\times A\) where \(A\) is an abelian finite group.NEWLINENEWLINEThis conjecture has been verified for many sporadic simple groups, but the authors of the present paper prove the conjecture for all the alternating groups of degree at least 5.NEWLINENEWLINEThis result extends the main result in [\textit{H. P. Tong-Viet}, Algebr. Represent. Theory 15, No. 2, 379--389 (2012; Zbl 1252.20005)]. Huppert himself proved the conjecture for the alternating groups of degrees up to 11, and for the degrees 12 and 13 it is proved in [\textit{H. N. Nguyen} et al., Algebra Colloq. 22, No. 2, 293--308 (2015; Zbl 1317.20005)].NEWLINENEWLINETherefore the authors in their proof assume the degree of the alternating group is at least 14.