所有非平凡自同构均无固定点的有限群

Finite Groups All of Whose Non-trivial Automorphisms Are Fixed-Point-Free

设G是一个有限群,证明了G的每个非平凡自同构均为无固定点自同构当且仅当G是阿贝尔单群.作为应用,证明了群G的全形是以G为Frobenius核的Frobenius群当且仅当G是奇素数阶群.

Let G be a finite group.It is proved that every non-trivial automorphism of G is fixed-point-free if and only if G is an abelian simple group.As an application,it is proved that the holomorph of G is a Frobenius group with G as its Frobenius kernel if and only if G is of odd prime order.