Inabstract algebra, aninner automorphism is anautomorphism of agroup,ring, oralgebra given by theconjugation action of a fixed element, called theconjugating element. They can be realized via operations from within the group itself, hence the adjective "inner". These inner automorphisms form asubgroup of the automorphism group, and thequotient of the automorphism group by this subgroup is defined as theouter automorphism group.

IfG is a group andg is an element ofG (alternatively, ifG is a ring, andg is aunit), then the function

is called(right) conjugation byg (see alsoconjugacy class). This function is anendomorphism ofG: for all${\displaystyle x_1,x_2\in G,}$

${\displaystyle {\phi }_g(x_1{x}_2)=g^{-1}{x}_1{x}_{2}g=g^{-1}{x}_1\left(g{g}^{-1}\right)x_{2}g=\left(g^{-1}{x}_{1}g\right)\left(g^{-1}{x}_{2}g\right)={\phi }_g(x_1){\phi }_g(x_2),}$

where the second equality is given by the insertion of the identity between${\displaystyle x_1}$ and${\displaystyle x_2.}$ Furthermore, it has a left and rightinverse, namely${\displaystyle {\phi }_{g^{-1}}.}$ Thus,${\displaystyle {\phi }_g}$ is both anmonomorphism andepimorphism, and so an isomorphism ofG with itself, i.e. an automorphism. Aninner automorphism is any automorphism that arises from conjugation.^[1]

When discussing right conjugation, the expression${\displaystyle g^{-1}xg}$ is often denoted exponentially by${\displaystyle x^g.}$ This notation is used because composition of conjugations satisfies the identity:${\displaystyle {\left(x^{g_1}\right)}^{g_2}=x^{g_1{g}_2}}$ for all${\displaystyle g_1,g_2\in G.}$ This shows that right conjugation gives a rightaction ofG on itself.

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

Describe a homomorphism${\displaystyle \mathrm{\Phi }}$ for which the image,${\displaystyle \text{Im}(\mathrm{\Phi })}$, is a normal subgroup of inner automorphisms of a group${\displaystyle G}$; alternatively, describe anatural homomorphism of which the kernel of${\displaystyle \mathrm{\Phi }}$ is the center of${\displaystyle G}$ (all${\displaystyle g\in G}$ for which conjugating by them returns the trivial automorphism), in other words,${\displaystyle \text{Ker}(\mathrm{\Phi })=\text{Z}(G)}$. There is always a natural homomorphism${\displaystyle \mathrm{\Phi }:G\to \text{Aut}(G)}$, which associates to every${\displaystyle g\in G}$ an (inner) automorphism${\displaystyle {\phi }_g}$ in${\displaystyle \text{Aut}(G)}$. Put identically,${\displaystyle \mathrm{\Phi }:g\mapsto {\phi }_g}$.

Let${\displaystyle {\phi }_g(x):=gx{g}^{-1}}$ as defined above. This requires demonstrating that (1)${\displaystyle {\phi }_g}$ is a homomorphism, (2)${\displaystyle {\phi }_g}$ is also abijection, (3)${\displaystyle \mathrm{\Phi }}$ is a homomorphism.

1. ${\displaystyle {\phi }_g(x{x}^{\prime })=gx{x}^{\prime }{g}^{-1}=gx(g^{-1}g)x^{\prime }{g}^{-1}=(gx{g}^{-1})(g{x}^{\prime }{g}^{-1})={\phi }_g(x){\phi }_g(x^{\prime })}$
2. The condition for bijectivity may be verified by simply presenting an inverse such that we can return to${\displaystyle x}$ from${\displaystyle gx{g}^{-1}}$. In this case it is conjugation by${\displaystyle g^{-1}}$denoted as${\displaystyle {\phi }_{g^{-1}}}$.
3. ${\displaystyle \mathrm{\Phi }(g{g}^{\prime })(x)=(g{g}^{\prime })x(g{g}^{\prime }{)}^{-1}}$ and${\displaystyle \mathrm{\Phi }(g)\circ \mathrm{\Phi }(g^{\prime })(x)=\mathrm{\Phi }(g)\circ (g^{\prime }x{g}^{\prime -1})=g{g}^{\prime }x{g}^{\prime -1}{g}^{-1}=(g{g}^{\prime })x(g{g}^{\prime }{)}^{-1}}$

Thecomposition of two inner automorphisms is again an inner automorphism, and with this operation, the collection of all inner automorphisms ofG is a group, the inner automorphism group ofG denotedInn(G).

Inn(G) is anormal subgroup of the fullautomorphism groupAut(G) ofG. Theouter automorphism group,Out(G) is thequotient group

${\displaystyle \mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G).}$

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

Saying that conjugation ofx bya leavesx unchanged is equivalent to saying thata andx commute:

${\displaystyle a^{-1}xa=x⟺xa=ax.}$

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

An automorphism of a groupG is inner if and only if it extends to every group containingG.^[4]

By associating the elementa ∈G with the inner automorphismf(x) =x^a inInn(G) as above, one obtains anisomorphism between thequotient groupG / Z(G) (whereZ(G) is thecenter ofG) and the inner automorphism group:

${\displaystyle G/\mathrm{Z}(G)\cong \mathrm{Inn}(G).}$

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

A result of Wolfgang Gaschütz says that ifG is a finite non-abelianp-group, thenG has an automorphism ofp-power order which is not inner.

It is anopen problem whether every non-abelianp-groupG has an automorphism of orderp. The latter question has positive answer wheneverG has one of the following conditions:

1. G is nilpotent of class 2
2. G is aregularp-group
3. G / Z(G) is apowerfulp-group
4. Thecentralizer inG,C_G, of the center,Z, of theFrattini subgroup,Φ, ofG,C_G ∘Z ∘ Φ(G), is not equal toΦ(G)

The inner automorphism group of a groupG,Inn(G), is trivial (i.e., consists only of theidentity element)if and only ifG isabelian.

The groupInn(G) iscyclic 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 calledcomplete. This is the case for all of the symmetric groups onn elements whenn is not 2 or 6. Whenn = 6, thesymmetric group has a unique non-trivial class of non-inner automorphisms, and whenn = 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 aperfect groupG is simple, thenG is calledquasisimple.

An automorphism of aLie algebra𝔊 is called an inner automorphism if it is of the formAd_g, whereAd is theadjoint map andg is an element of aLie group whose Lie algebra is𝔊. 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.

IfG is thegroup of units of aring,A, then an inner automorphism onG can be extended to a mapping on theprojective line overA by the group of units of thematrix ring,M₂(A). In particular, the inner automorphisms of theclassical groups can be extended in that way.

• Abdollahi, A. (2010), "Powerfulp-groups have non-inner automorphisms of orderp and some cohomology",J. Algebra,323 (3):779–789,arXiv:0901.3182,doi:10.1016/j.jalgebra.2009.10.013,MR 2574864
• Abdollahi, A. (2007), "Finitep-groups of class2 have noninner automorphisms of orderp",J. Algebra,312 (2):876–879,arXiv:math/0608581,doi:10.1016/j.jalgebra.2006.08.036,MR 2333188
• Deaconescu, M.; Silberberg, G. (2002), "Noninner automorphisms of orderp of finitep-groups",J. Algebra,250:283–287,doi:10.1006/jabr.2001.9093,MR 1898386
• Gaschütz, W. (1966), "Nichtabelschep-Gruppen besitzen äusserep-Automorphismen",J. Algebra,4:1–2,doi:10.1016/0021-8693(66)90045-7,MR 0193144
• Liebeck, H. (1965), "Outer automorphisms in nilpotentp-groups of class2",J. London Math. Soc.,40:268–275,doi:10.1112/jlms/s1-40.1.268,MR 0173708
• Remeslennikov, V.N. (2001) [1994],"Inner automorphism",Encyclopedia of Mathematics,EMS Press
• Weisstein, Eric W."Inner Automorphism".MathWorld.

• Group theory
• Group automorphisms

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