Inner automorphism

Short description: 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.

Definition

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

[math]\displaystyle{ \begin{align} \varphi_g\colon G&\to G \\ \varphi_g(x)&:= g^{-1}xg \end{align} }[/math]

is called (right) conjugation by g (see also conjugacy class). This function is an endomorphism of G: for all [math]\displaystyle{ x_1,x_2\in G, }[/math]

[math]\displaystyle{ \varphi_g(x_1 x_2) = g^{-1} x_1 x_2g = 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) = \varphi_g(x_1)\varphi_g(x_2), }[/math]

where the second equality is given by the insertion of the identity between [math]\displaystyle{ x_1 }[/math] and [math]\displaystyle{ x_2. }[/math] Furthermore, it has a left and right inverse, namely [math]\displaystyle{ \varphi_{g^{-1}}. }[/math] Thus, [math]\displaystyle{ \varphi_g }[/math] is bijective, and so an isomorphism of G with itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation.

When discussing right conjugation, the expression [math]\displaystyle{ g^{-1}xg }[/math] is often denoted exponentially by [math]\displaystyle{ x^g. }[/math] This notation is used because composition of conjugations satisfies the identity: [math]\displaystyle{ \left(x^{g_1}\right)^{g_2} = x^{g_1g_2} }[/math] for all [math]\displaystyle{ g_1, g_2 \in G. }[/math] This shows that right conjugation gives a right action of G on itself.

Inner and outer automorphism groups

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

[math]\displaystyle{ \operatorname{Out}(G) = \operatorname{Aut}(G) / \operatorname{Inn}(G). }[/math]

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:

[math]\displaystyle{ a^{-1}xa = x \iff xa = ax. }[/math]

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.

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:

[math]\displaystyle{ G\,/\,\mathrm{Z}(G) \cong \operatorname{Inn}(G). }[/math]

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).

Non-inner automorphisms of finite p-groups

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:

1. G is nilpotent of class 2
2. G is a regular p-group
3. G / Z(G) is a powerful p-group
4. 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)

Types of groups

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.

Lie algebra case

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.

Extension

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, M₂(A). In particular, the inner automorphisms of the classical groups can be extended in that way.

Further reading

• Abdollahi, A. (2010), "Powerful p-groups have non-inner automorphisms of order p and some cohomology", J. Algebra 323 (3): 779–789, doi:10.1016/j.jalgebra.2009.10.013 
• Abdollahi, A. (2007), "Finite p-groups of class 2 have noninner automorphisms of order p", J. Algebra 312 (2): 876–879, doi:10.1016/j.jalgebra.2006.08.036 
• Deaconescu, M.; Silberberg, G. (2002), "Noninner automorphisms of order p of finite p-groups", J. Algebra 250: 283–287, doi:10.1006/jabr.2001.9093 
• Gaschütz, W. (1966), "Nichtabelsche p-Gruppen besitzen äussere p-Automorphismen", J. Algebra 4: 1–2, doi:10.1016/0021-8693(66)90045-7 
• Liebeck, H. (1965), "Outer automorphisms in nilpotent p-groups of class 2", J. London Math. Soc. 40: 268–275, doi:10.1112/jlms/s1-40.1.268 
• Hazewinkel, Michiel, ed. (2001), "Inner automorphism", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=I/i051230 
• Weisstein, Eric W.. "Inner Automorphism". http://mathworld.wolfram.com/InnerAutomorphism.html. 

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