Inner automorphism

In abstract algebra an inner automorphism is a certain type of automorphism of a group defined in terms of a fixed element of the group, called the conjugating element. Formally, if G is a group and a is an element of G, then the inner automorphism defined by a is the map f from G to itself defined for all x in G by the formula

f(x) = a⁻¹xa.

Here we use the convention that group elements act on the right.

The operation x ↦ a⁻¹xa is called conjugation (see also conjugacy class), and it is often of interest to distinguish the cases where conjugation by one element leaves another element unchanged from cases where conjugation generates a new element.

In fact, saying that conjugation of x by a leaves x unchanged is equivalent to saying that a and x commute:

a⁻¹xa = x ⇔ ax = xa.

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.

An automorphism of a group G is inner if and only if it extends to every group containing G.^[1]

Notation

The expression a⁻¹xa is often denoted exponentially by x^a. This notation is used because we have the rule (x^a)^b = x^ab (giving a right action of G on itself).

Properties

Every inner automorphism is indeed an automorphism of the group G, i.e. it is a bijective map from G to G and it is a homomorphism; meaning that (xy)^a = x^ay^a.

Inner and outer automorphism groups

The composition of two inner automorphisms is again an inner automorphism (as mentioned above: (x^a)^b = x^ab), and with this operation, the collection of all inner automorphisms of G is itself 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

Out(G) ≡ Aut(G)/Inn(G)

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).

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 centre of G) and the inner automorphism group:

G/Z(G) = Inn(G).

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 centre, Z, of the Frattini subgroup, Φ, of G, C_G∘Z∘Φ(G), is not equal to Φ(G)

Types of groups

It follows that the inner automorphism group, Inn(G), is trivial (i.e., consists only of the identity element) if and only if G is abelian.

It is easy to prove that Inn(G) can be a cyclic group 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 centre 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 outer automorphisms, and when n = 2 the symmetric group, despite having no outer automorphisms, is abelian, giving a non-trivial centre disqualifying it from being complete.

If the inner automorphism group of a perfect group G is simple, then G is called quasisimple.

Ring case

Given a ring, R, and a unit, u in R, the map f(x) = u⁻¹xu is a ring automorphism of R. The ring automorphisms of this form are called inner automorphisms of R. They form a normal subgroup of the automorphism group of R.

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.

References

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• Abdollahi, A. (2010), "Powerful p-groups have non-inner automorphisms of order p and some cohomology", J. Algebra, 323: 779–789, doi:10.1016/j.jalgebra.2009.10.013, MR 2574864
• Abdollahi, A. (2007), "Finite p-groups of class 2 have noninner automorphisms of order p", J. Algebra, 312: 876–879, doi:10.1016/j.jalgebra.2006.08.036, MR 2333188
• 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, MR 1898386
• Gaschütz, W. (1966), "Nichtabelsche p-Gruppen besitzen äussere p-Automorphismen", J. Algebra, 4: 1–2, doi:10.1016/0021-8693(66)90045-7, MR 0193144
• Liebeck, H. (1965), "Outer automorphisms in nilpotent p-groups of class 2", J. London Math. Soc., 40: 268–275, MR 0173708
• Remeslennikov, V.N. (2001), "Inner automorphism", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Weisstein, Eric W. "Inner Automorphism". MathWorld.