===Definition with symbols===
The '''automorphism group''' of a [[group]] <math>G</math>, denoted <math>Aut(G)</math>, is a set whose elements are automorphisms <math>\sigma:G \to G</math>, and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of <math>Sym(G)</math>, the group of all permutations on <math>G</math>.
Every [[group-closed automorphism property]] gives rise to a [[normal subgroup]] of the automorphism group.
Examples are the property of being an [[ inner automorphism]], [[ class automorphism]], [[ extensible automorphism]].