Let G be a finite group such that Aut(G)={e}. Prove that G is isomorphic to ℤ₂

Since Aut(G)={e} we would have that G is abelian (because we would have G/Z(G) ={e} and so G=Z(G) ) then the inversion x|—>x⁻¹ is an automorphism of G. If G has an element of order greater then 2 then the inversion is not trivial so G is a 2-elementary abelian groups and so G is a ℤ₂ – vector space of dimension n. Then Aut(G)=GL(n,ℤ₂). If n>1 the general linear group GL(n,ℤ₂) is not trivial so n=1. Then G=ℤ₂. ■