Inner automorphism

In abstract algebra an inner automorphism is a function which, informally, involves a certain operation being applied, then another one (x) performed, and then the initial operation being reversed. Sometimes this has a net effect ("take off shoes, take off socks, put on shoes"), and sometimes it does not ("take off left glove, take off right glove, put on left glove" or "take off right glove" are equivalent).

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

ƒ: G → G

defined by

ƒ(x) = a⁻¹xa, for all x in G,

where a is a given fixed element of G.

The operation a⁻¹xa is called conjugation (see also conjugacy class).

In fact

a⁻¹xa = x

is equivalent to saying

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.

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 (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 quotient group

Aut(G)/Inn(G)

is known as the outer automorphism group Out(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 in G with the inner automorphism ƒ(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:

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. The centralizer C_G(Z(Φ(G))) in G of the center of the Frattini subgroup Φ(G) of G is not equal to Φ(G)
4. G/Z(G) is a powerful p-group

Types of groups

It follows that the group Inn(G) of inner automorphisms is itself 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 center of a group.

At the opposite end of the spectrum, it is possible that the inner automorphisms exhaust the entire automorphism group; a group whose automorphisms are all inner is called 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 ƒ(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 arises as the group of units of a ring A, then an inner automorphism on G can be extended to a projectivity on the projective space over A by inversive ring geometry. In particular, the inner automorphisms of the classical linear groups can be so extended.

References

• Abdollahi, A. (2010), "Powerful p-groups have non-inner automorphisms of order p and some cohomology", J. Algebra 323: 779–789, MR2574864 
• Abdollahi, A. (2007), "Finite p-groups of class 2 have noninner automorphisms of order p", J. Algebra 312: 876–879, MR2333188 
• Deaconescu, M.; Silberberg, G. (2002), "Noninner automorphisms of order p of finite p-groups", J. Algebra 250: 283–287, MR1898386 
• Gaschütz, W. (1966), "Nichtabelsche p-Gruppen besitzen äussere p-Automorphismen", J. Algebra 4: 1–2, MR0193144 
• Liebeck, H. (1965), "Outer automorphisms in nilpotent p-groups of class 2", J. London Math. Soc. 40: 268–275, MR0173708 
• Remeslennikov, V.N. (2001), "Inner automorphism", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/I/i051230.htm 
• Weisstein, Eric W., "Inner Automorphism" from MathWorld.