On characterization by order and prime graph for alternating groups.

There are some graphs related to the structure of a finite group, e.g. the prime graph, the solvable graph, and the noncommuting graph of a finite group. In this paper, we consider characterization by order and prime graph for alternating groups. In particular, we show that there exists a closed relation between characterization by order and prime graph of some of the alternating groups and Goldbach's conjecture. Also by using the main results, we answer some conjectures and a problem about characterization by order and degree pattern for alternating and symmetric groups that are stated in [R. Kogani-Moghaddam and A. R. Moghaddamfar, 'Groups with the same order and degree pattern,' Sci. China Math., 2012]. [ABSTRACT FROM AUTHOR]