In abstract algebra, an inner automorphism of a group is a function f : G -> G defined by Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

f(x) = axa^-1

for all x in G; where the conjugation is often denoted exponentially by ^ax. This notation is used because we have the rule ^a(^bx)=^abx (giving a left action of G on itself). An alternative form, leading to a right action, can be obtained by (re)defining f(x) to be a^-1xa; this form is denoted x^a

As the name suggests, f is an automorphism of G. An automorphism not of this form is called an outer automorphism. In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...

The collection of all inner automorphisms of G is a group, denoted Inn(G). It is a normal subgroup of the full automorphism group Aut(G) of G. The quotient group Aut(G)/Inn(G) is known as the outer automorphism group Out(G) (the elements of that group are cosets of automorphisms, and hence are not actually the outer automorphisms, since those can't form a group.) In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g-1ng is still in N. The statement N is a normal subgroup of G is written: . Another way... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ...

By associating the element a in G with the inner automorphism f in Inn(G) as above, one obtains an isomorphism between the factor group G/Z(G) (where Z(G) is the center of G) and Inn(G). As a consequence, the group Inn(G) of inner automorphisms is trivial (i.e. consists only of the identity element) if and only if G is abelian. In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... In abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of G — if... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, an abelian group is a commutative group, i. ...

At the opposite end of the spectrum, it is possible that the inner automorphisms exhaust the entire automorphism group---a group whose automorphisms are all inner is called complete. In mathematics, a group G is said to be complete if all the automorphisms of G are inner. ...

An automorphism of a Lie algebra is called an inner automorphism if it is of the form Ad_g, where Ad is the adjoint map and g is an element of a Lie group whose Lie algebra is . 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. In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, the adjoint endomorphism or adjoint action is an endomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras and Lie groups. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...