Let $(G,\cdot)$ be a group.
If $G$ is Abelian, the commutator subgroup $[G,G]=\{ghg^{-1}h^{-1}|g,h\in G\}$ is trivial; otherwise, the commutator subgroup is not trivial and there is the Abelianisation $\mathrm{Ab}(G)=G/[G,G]$.
Also, if $G$ is Abelian, then the centre is itself or $\mathrm{Z}(G)=G$. Thus, its Inner Automorphism group $\mathrm{Inn}(G)\cong G/\mathrm{Z}(G)$ is trivial.
I can see that if $G$ is not Abelian, then its Inner Automorphism group will not be trivial, but I do not know what happens next.
Are there some interesting relationships between $[G,G]$ and $\mathrm{Inn}(G)$? Still, why are commutators and inner automorphisms both about commutativity of one group?
Thanks.
Best Answer
If $[G,G]$ is trivial then $ghg^{-1}h^{-1}=e$ and so $ghg^{-1}=h$. So for every $g$ and $h$ we see that $h$ is stabilized by conjugation. Said another way, the group of inner automorphisms is trivial.