Question

An Introduction of the Theory of Groups - Fourth Edition (Joseph J. Rotman)

If G is a finite group and Aut(G) acts transitively on G^# = G − 1, then prove that G is an elementary abelian group.

Answer 1

a. associate degree automorphism maps the middle of G to the center of G. Thus, below your condition the middle of G is that the full cluster and it's so abelian, or the middle is trivial.

b. Note that associate degree automorphism preserves the order of a part. so the transitivity implies that each one component of G except one have the identical order n. And this n (thus) needs to be prime. particularly, the middle is non-trivial.

Thus, G may be a finite abelian p-group during which all components (except 1) have order n and n is prime. That is, it's associate degree elementary abelian p-group.