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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

where the second equality is given by the insertion of the identity between and Furthermore, it has a left and right inverse, namely Thus, is both an monomorphism and epimorpism, and so an isomorphism of G with itself, i.e. an automorphism. An **inner automorphism** is any automorphism that arises from conjugation.^{[1]}

When discussing right conjugation, the expression is often denoted exponentially by This notation is used because composition of conjugations satisfies the identity: for all This shows that right conjugation gives a right action of G on itself.

A common example is as follows:^{[2]}^{[3]}

Describe a homomorphism for which the image, , is a normal subgroup of inner automorphisms of a group ; alternatively, describe a natural homomorphism of which the kernel of is the center of (all for which conjugating by them returns the trivial automorphism), in other words, . There is always a natural homomorphism , which associates to every an (inner) automorphism in . Put identically, .

Let as defined above. This requires demonstrating that (1) is a homomorphism, (2) is also a bijection, (3) is a homomorphism.

- The condition for bijectivity may be verified by simply presenting an inverse such that we can return to from . In this case it is conjugation by denoted as .
- and

The composition of two inner automorphisms is again an inner automorphism, and with this operation, the collection of all inner automorphisms of G is a group, the inner automorphism group of G denoted Inn(*G*).

Inn(*G*) is a normal subgroup of the full automorphism group Aut(*G*) of G. The outer automorphism group, Out(*G*) is the quotient group

The outer automorphism group measures, in a sense, how many automorphisms of G are not inner. Every non-inner automorphism yields a non-trivial element of Out(*G*), but different non-inner automorphisms may yield the same element of Out(*G*).

Saying that conjugation of x by a leaves x unchanged is equivalent to saying that a and x commute:

Therefore the existence and number of inner automorphisms that are not the identity mapping is a kind of measure of the failure of the commutative law in the group (or ring).

An automorphism of a group G is inner if and only if it extends to every group containing G.^{[4]}

By associating the element *a* ∈ *G* with the inner automorphism *f*(*x*) = *x*^{a} in Inn(*G*) as above, one obtains an isomorphism between the quotient group *G* / Z(*G*) (where Z(*G*) is the center of G) and the inner automorphism group:

This is a consequence of the first isomorphism theorem, because Z(*G*) is precisely the set of those elements of G that give the identity mapping as corresponding inner automorphism (conjugation changes nothing).

A result of Wolfgang Gaschütz says that if G is a finite non-abelian p-group, then G has an automorphism of p-power order which is not inner.

It is an open problem whether every non-abelian p-group G has an automorphism of order p. The latter question has positive answer whenever G has one of the following conditions:

- G is nilpotent of class 2
- G is a regular p-group
*G*/ Z(*G*) is a powerful p-group- The centralizer in G,
*C*_{G}, of the center, Z, of the Frattini subgroup, Φ, of G,*C*_{G}∘*Z*∘ Φ(*G*), is not equal to Φ(*G*)

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 *n* = 6, the symmetric group has a unique non-trivial class of non-inner automorphisms, and when *n* = 2, 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 Ad_{g}, 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, M_{2}(*A*). In particular, the inner automorphisms of the classical groups can be extended in that way.

**^**Dummit, David S.; Foote, Richard M. (2004).*Abstract algebra*(3rd ed.). Hoboken, NJ: Wiley. p. 45. ISBN 978-0-4714-5234-8. OCLC 248917264.**^**Grillet, Pierre (2010).*Abstract Algebra*(2nd ed.). New York: Springer. p. 56. ISBN 978-1-4419-2450-6.**^**Lang, Serge (2002).*Algebra*(3rd ed.). New York: Springer-Verlag. p. 26. ISBN 978-0-387-95385-4.**^**Schupp, Paul E. (1987), "A characterization of inner automorphisms" (PDF),*Proceedings of the American Mathematical Society*,**101**(2), American Mathematical Society: 226–228, doi:10.2307/2045986, JSTOR 2045986, MR 0902532

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