Center of a group

That is to say $(G, *)$ a group, noted multiplicativement, of neutral $E$ .

Definition

One calls center of the group $G$ the whole of the elements which commutate with all the others:

Z_G = \ left \ {Z \ in G/\ forall G \ in G, G Z = Z G \ right \}

$Z_G$ is a Sous-groupe of $G$ .

One shows that $Z_G$ is a Sous-groupe distinguished, Abélien.

The center of an abelian group $G$ is the whole $G$ group, i.e.:

$Z_G = G \,$

One considers the interior Automorphisme:

\ phi: G \ rightarrow Aut (G), \, G \ mapsto \ phi_g \,

where $\ phi_g \,$ is the automorphism defined by:

\ phi _g: G \ rightarrow G, H \ mapsto G H g^ {- 1} \,

One has then:

\ ker (\ phi) =Z_G \,

\ mbox {Im} \, G= \ mbox {Inn} (G)

The sub-group $\ mbox {Inn} (G)$ is called group of the interior automorphisms of G.

One can deduce some, according to the theorem of isomorphism:

$G/Z (G) \ cong \ mbox {Inn} (G) \,$ .

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