Inner automorphism

In abstract algebra an inner automorphism is a function in which an initial operation is applied, then another operation, and then the initial operation is reversed. With letters to indicate the operations and thing-being-transformed: f⁻¹∘g∘f(X). Sometimes the initial action and its subsequent reversal change the overall result ("raise umbrella, walk through rain, lower umbrella" has a different result from just "walk through rain"), and sometimes they do not ("take off left glove, take off right glove, put on left glove" has the same effect as only "take off right glove").

More formally an inner automorphism of a group G is a function

f : G → G

defined for all x in G by

f(x) = a⁻¹xa,

where a is a given fixed element of G, and where we deem the action of group elements to occur on the right.

The operation 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 (as in the "gloves" analogy above) from cases where conjugation generates a new element (as in the "umbrella" analogy).

In fact, saying that conjugation of x by a leaves x unchanged, i.e.

a⁻¹xa = x,

is equivalent to saying that a and x commute:

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 $\mathrm{Inn}(G)$.

$\mathrm{Inn}(G)$ is a normal subgroup of the full automorphism group $\mathrm{Aut}(G)$ of $G$. The outer automorphism group, $\mathrm{Out}(G)$ is the quotient group

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

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 $\mathrm{Out}(G)$, but different non-inner automorphisms may yield the same element of $\mathrm{Out}(G)$.

By associating the element $a \in G$ with the inner automorphism $f(x)=x^a$ in $\mathrm{Inn}(G)$ 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:

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

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.

Inn(G) can only be a cyclic group when it is trivial, by a basic result on the centre of a group.

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 $\mathfrak{g}$ 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 $\mathfrak{g}$. 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.