Automorphism group ($\mathrm{Aut}(G)$)

The automorphism group of a group $G$ is the group of all isomorphisms $G\to G$ under composition.

Write the finite cyclic group as $C_n=⟨r⟩$. A group homomorphism is determined by $r\mapsto r^a$; it is a group automorphism exactly when $r^a$ generates $C_n$, equivalently $\gcd (a,n)=1$. Composition multiplies the exponents modulo $n$, proving $\mathrm{Aut}(C_n)\cong (\mathbb{Z}/n\mathbb{Z}{)}^{\times }$. Its group order is therefore Euler's totient function $\varphi (n)$. A group homomorphism $C_2\to \mathrm{Aut}(C_n)$ is determined by a unit $a$ with $a^2\equiv 1(\mathrm{mod}n)$.

The holomorph acts on the underlying set of a group by $x\mapsto g\alpha (x)$. Its multiplication is $(g,\alpha )(h,\beta )=(g\alpha (h),\alpha \beta )$. This is exactly the normalizer of the left regular permutation representation: a normalizing permutation fixing the identity is an group automorphism, and every normalizing permutation is a left translation followed by one. The embedded copy of $G$ is normal and the identity stabilizer is its group automorphism group.

An inner automorphism is conjugation by an element of the group. The inner automorphisms form a normal subgroup $\mathrm{Inn}(G)◃\mathrm{Aut}(G)$.

The outer automorphism group isIt records automorphisms modulo conjugation by elements of $G$.

An outer group automorphism is not conjugation by any element of the group. Its nonidentity class belongs to $\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G)$. A permutation-group example is the exceptional automorphism of $S_6$ that swaps its natural and transitive copies of $S_5$; it sends a transposition to a product of three disjoint transpositions and hence cannot be inner.