On Huppert's Conjecture for alternating groups of low degrees

Bertram Huppert conjectured in the late 1990s that the nonabelian simple groups are determined up to an abelian direct factor by the set of their character degrees. Although the conjecture has been established for various simple groups of Lie type and simple sporadic groups, it is expected to be difficult for alternating groups. In [5], Huppert verified the conjecture for the simple alternating groups An of degree up to 11. In this paper, we continue his work and verify the conjecture for the alternating groups of degrees 12 and 13