I am interested in a classification of groups with trivial automorphism group. For the purposes of this question I will refer to such groups as asymmetric because they lack symmetry (in particular they lack non trivial symmetry). If there is a correct name for these groups please let me know and feel free to update my question. Here is my attempt to classify asymmetric groups:
First, notice that the trivial group and the group $Z_2$ both an examples of such a group. Next notice any such group must be abelian as shown below.
Say $G$ is asymmetric, then for each automorphism $ø$ we have that $ø(x)=x$. Consider the automorphism $ø_y(x) = y^{-1}xy$. Because $ø_y(x)=x$ we have $x=y^{-1}xy$ thus $yx=xy$. Thus $\forall x,y \in G, xy=yx$. Thus G is abelian.
Next notice the order of any non identity element in G must be 2 as shown below.
Consider the function $ø(x)=x^{-1}$. This is an automorphism because $ø$ is bijective and $ø(xy)=(xy)^{-1}=y^{-1}x^{-1}=ø(y)ø(x)=ø(x)ø(y)$ where the last equality holds because G is abelian as shown earlier. Recall $ø(x)=x$ because G is asymmetric, thus $x=ø(x)=x^{-1}$ thus $x^2=e$ where $e$ is the identity element. Thus $\forall x\neq e, ord(x)=2$.
From these two results and the classification of finite abelian groups it is clear that $Z_2$ and the trivial group are the only finite asymmetric groups however I have been unable to prove there are no asymmetric infinite groups. If you have any idea how to do this please let me know. Thanks for your time,
Mathew
