Inner automorphism

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In abstract algebra, an inner automorphism of a group G is a function

f : G → G

defined by

f(x) = axa⁻¹, where a is a given fixed element of G,

for all x in G.

The operation axa⁻¹ is called conjugation (see also conjugacy class). Informally a certain operation is applied, then x follows, and the initial operation is reversed ('take off shoes, take off socks, replace shoes'). Sometimes this matters, and sometimes ('take off left glove, take off right glove, replace left glove') it doesn't.

Contents 1 Structure 2 Notation 3 The inner automorphism group 4 Lie algebra case 5 Extension 6 Formal proofs about Inn(G)

[edit] Structure

The inner automorphisms of G are therefore the set of all homomorphisms of G to itself, such that the action is conjugation by an element g in G. The inner automophisms, denoted by

Inn(G),

themselves form a group, which is a subgroup of the automorphism group of G, written Aut(G). An automorphism of G not of this form may be called an outer automorphism, but this terminology is a little imprecise, as will be explained below.

[edit] Notation

It is often denoted exponentially by ^ax. This notation is used because we have the rule ^a(^bx)=^abx (giving a left action of G on itself). An alternative form, leading to a right action, can be obtained by (re)defining f(x) to be a⁻¹xa; this form is denoted x^a.

[edit] The inner automorphism group

The collection of all inner automorphisms of G is a group, denoted Inn(G). It 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). Despite the name, the elements of Out(G) are not outer automorphisms, but cosets of automorphisms; the outer automorphisms do not form a group.

By associating the element a in G with the inner automorphism f in Inn(G) as above, one obtains an isomorphism between the factor group

G/Z(G),

where Z(G) is the center of G. This is because Z(G) is precisely the subset of G giving the identity mapping as corresponding inner automorphism (conjugation changes nothing). As a consequence, 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.

[edit] Lie algebra case

An automorphism of a Lie algebra 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 . 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.

[edit] 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.

[edit] Formal proofs about Inn(G)

Inn(G) is a normal subgroup of Aut(G).

Proof:

Let α ∈ Aut(G), i_x ∈ Inn(G)

αi_xgα^-1 = αgx(αg)^-1 = i_α(g) = i_g' s.t. g'=α(g) Q.E.D. 

G/Z(G) is isomorphic with Inn(G).

Proof:

Let

φ: G→Inn(G)

such that

g→gxg^-1

then

ker(φ)={x ∈ G | gxg^-1=x ∀ g ∈ G}.

So, by definition of the center of G,

ker(φ)=Z(G).

Therefore, by the first isomorphism theorem, G/Z(G)≈Inn(G) Q.E.D.