# Defining an inner automorphism

Trying to properly understand inner automorphisms. So as far as the definition goes, an automorphism of a group is termed an inner automorphism if it can be expressed as conjugation by an element of the group.

Or for an alternate definition an automorphism of a group G is considered an inner automorphism if there is an element $g$ in $G$ such that $\forall x \in G, c_g(x) = gxg^{-1}$.

$Inn(G)$ is simply the group of all such automorphisms as defined above.

Now my question lies with identifying whether something is an inner automorphism or not. Is it correct to simply say that if a function isn't expressed in that form, we can conclude that it isn't an inner automorphism.

• An automorphism may be inner in a non-obvious way, so no. You need to do some work to show that there is no way to write your automorphism as an inner automorphism. – Qiaochu Yuan Nov 17 '17 at 4:57

As an example, consider the duality automorphism of the special linear group ${\rm SL}(n,K)$ over a field $K$ that maps each matrix $A$ to the transpose ${\rm Tr}(A^{-1})$ of $A^{-1}$.
Is this an inner automrphism? The answer turns out to be yes if $n=2$, but no when $n > 2$, but that is not at al obvious. The corresponding automorphism of ${\rm GL}(2,K)$ is not inner when $|K|>2$.