Inner automorphism

In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element. They can be realized via operations from within the group itself, hence the adjective "inner". These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup is defined as the outer automorphism group.

If G is a group and g is an element of G (alternatively, if G is a ring, and g is a unit), then the function

is called (right) conjugation by g (see also conjugacy class). This function is an endomorphism of G: for all ${\displaystyle x_{1},x_{2}\in G,}$

where the second equality is given by the insertion of the identity between ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}.}$ Furthermore, it has a left and right inverse, namely ${\displaystyle \varphi _{g^{-1}}.}$ Thus, ${\displaystyle \varphi _{g}}$ is both an monomorphism and epimorphism, and so an isomorphism of G with itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation.

When discussing right conjugation, the expression ${\displaystyle g^{-1}xg}$ is often denoted exponentially by ${\displaystyle x^{g}.}$ This notation is used because composition of conjugations satisfies the identity: ${\displaystyle \left(x^{g_{1}}\right)^{g_{2}}=x^{g_{1}g_{2}}}$ for all ${\displaystyle g_{1},g_{2}\in G.}$ This shows that right conjugation gives a right action of G on itself.

A common example is as follows:

Describe a homomorphism ${\displaystyle \Phi }$ for which the image, ${\displaystyle {\text{Im}}(\Phi )}$, is a normal subgroup of inner automorphisms of a group ${\displaystyle G}$; alternatively, describe a natural homomorphism of which the kernel of ${\displaystyle \Phi }$ is the center of ${\displaystyle G}$ (all ${\displaystyle g\in G}$ for which conjugating by them returns the trivial automorphism), in other words, ${\displaystyle {\text{Ker}}(\Phi )={\text{Z}}(G)}$. There is always a natural homomorphism ${\displaystyle \Phi :G\to {\text{Aut}}(G)}$, which associates to every ${\displaystyle g\in G}$ an (inner) automorphism ${\displaystyle \varphi _{g}}$ in ${\displaystyle {\text{Aut}}(G)}$. Put identically, ${\displaystyle \Phi :g\mapsto \varphi _{g}}$.

Let ${\displaystyle \varphi _{g}(x):=gxg^{-1}}$ as defined above. This requires demonstrating that (1) ${\displaystyle \varphi _{g}}$ is a homomorphism, (2) ${\displaystyle \varphi _{g}}$ is also a bijection, (3) ${\displaystyle \Phi }$ is a homomorphism.