Interior automorphism

A interior automorphism is a notion Mathématique used in Théorie of the groups.

That is to say G a group and G an element of G .

One calls interior automorphism associated with G , noted ι_g, the automorphism defined by:

\ forall X \ in G \ quad \ iota_g (X) =g.x.g^ {- 1} \;

For a commutative group, the interior automorphisms are commonplace. More generally, the whole of the interior automorphisms of G form a normal Sous-groupe Groupe of the automorphisms of G , isomorph with the Quotient of G by its center. Isomorphism is induced by the Action by conjugation of G on itself.

Definitions

Interior automorphism

* Is G a group, the application of G in G ι is known as interior automorphism if and only if the following property is checked:

\ exists G \ in G \ quad \ forall X \ in G \ quad \ iota (X) =gxg^ {- 1} \;

One speaks then about interior automorphism by G , and one uses sometimes the notation ι_g.

It is noticed that an interior automorphism is a bijective morphism , indeed:

\ forall X, there \ in G \ quad \ iota_g (xy) =gxyg^ {- 1} = (gxg^ {- 1}). (gyg^ {- 1}) = \ iota_g (X). \ iota_g (there) \;

A calculation quite as direct gives:

\ iota_ {gh} = \ iota_g \ circ \ iota_h \;

In particular, ι_g is an automorphism of the group G , whose reverse is ι_g^-1.

If G is an central element of G (IE. an element of the center Z ( G ) of G ), the interior automorphism by G is the identity. More generally, the whole of the fixed points of ι_g is exactly the Centralisateur of G .

* If X and is there two elements of G such as X is the image of there by an interior automorphism, then X and is there known as combined .

Note: If G is provided with additional structures (topological Groupe, Groupe of Dregs, algebraic Groupe), the interior automorphisms are always isomorphisms for the structures considered.

Sub-group normal

Group interior automorphisms

The application $\ iota: G \ mapsto \ iota_g$ is a morphism of groups of G in the group $Aut (G)$ of the automorphisms of G . The image is exactly the whole of the interior automorphisms of G , which is thus a sub-group of $Aut (G)$ , noted $Int (G)$ . By the Lemma of factorization, the surjective morphism $\ iota: G \ rightarrow Int (G)$ induces an isomorphism:

G/Z (G) \ rightarrow Int (G)

If $\ phi$ is an automorphism of G , and if G is an element of G , a calculation gives:

\ forall X \ in G, \, \ phi \ iota_g \ phi^ {- 1} (X) = \ phi \ left= \ phi (G) .x. \ phi (G) ^ {- 1}

, from where:

\ phi \ iota_ {G} \ phi^ {- 1} = \ iota_ {\ phi (G)}

. Combined of an interior automorphism by an automorphism is thus an interior automorphism. In fact,

Int (G)

is a normal Sous-groupe of

Aut (G)

. To summarize, one thus has the following exact continuation:

1 \ rightarrow Z (G) \ rightarrow G \ rightarrow Int (G) \ rightarrow Aut (G) \ rightarrow Aut (G) /Int (G) \ rightarrow 1

Group automorphism of a sub-group normal

With the notations above, if H is a sub-group normal of G , any interior automorphism of G is restricted in an automorphism of H . From where a possibly surjective morphism of groups $Int (G) \ rightarrow Aut (H)$ . The surjectivity is hoped for to determine the group of the automorphisms of H .

The composition by $\ iota$ gives a morphism $G \ rightarrow Aut (H)$ , whose core is the switch of H .

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