Группаның, сақинаның немесе алгебраның ішкі автоморфизмі

Automorphism of a group, ring, or algebra given by the conjugation action of one of its elements
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 simple 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.

The inner automorphism group of a group G, Inn(G), is trivial (i. e., consists only of the identity element) if and only if G is abelian. The group Inn(G) is cyclic only when it is trivial. At the opposite end of the spectrum, the inner automorphisms may exhaust the entire automorphism group; a group whose automorphisms are all inner and whose center is trivial is called complete. This is the case for all of the symmetric groups on n elements when n is not 2 or 6. When , the symmetric group has a unique non trivial class of non inner automorphisms, and when , the symmetric group, despite having no non inner automorphisms, is abelian, giving a non trivial center, disqualifying it from being complete. If the inner automorphism group of a perfect group G is simple, then G is called quasisimple.

An automorphism of a Lie algebra 𝔊 is called an inner automorphism if it is of the form , where Ad is the adjoint map and g is an element of a Lie group whose Lie algebra is 𝔊. The notion of inner automorphism for Lie algebras is compatible with the notion for groups in the sense that an inner automorphism of a Lie group induces a unique inner automorphism of the corresponding Lie algebra.

If G is the group of units of a ring, A, then an inner automorphism on G can be extended to a mapping on the projective line over A by the group of units of the matrix ring, In particular, the inner automorphisms of the classical groups can be extended in that way.