Mutations in finite groups.
In the present paper, we classify groups with the same order and degree pattern as an almost simple group related to the projective special linear simple group . As a consequence of this result we can give a positive answer to a conjecture of W. J. Shi and J. X. Bi, for all almost simple groups related to except . Also, we prove that if is an almost simple group related to except and is a finite group such that and , then .
We discuss a group-theoretical generalization of the well-known Gauss formula involving the function that counts the number of automorphisms of a finite group. This gives several characterizations of finite cyclic groups.