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Difference between revisions of "Normal subgroup"

From Online Dictionary of Crystallography

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(Added German and Spanish translations (U. Mueller))
 
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<font color="blue"> Sous-groupe normal </font> (''Fr''). <font color="black"> Sottogruppo normale </font> (''It''). <font color="purple"> 正規部分群 </font> (''Ja'').
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<font color="blue">Sous-groupe normal</font> (''Fr''). <font color="red">Normalteiler</font> (''Ge''). <font color="black">Sottogruppo normale</font> (''It''). <font color="purple">正規部分群</font> (''Ja''). <font color="green">Subgrupo normal</font> (''Sp'').
  
 
== Definition ==
 
== Definition ==

Latest revision as of 13:14, 16 November 2017

Sous-groupe normal (Fr). Normalteiler (Ge). Sottogruppo normale (It). 正規部分群 (Ja). Subgrupo normal (Sp).

Definition

A subgroup H of a group G is normal in G (H [math]\triangleleft[/math] G) if gH = Hg for any gG. Equivalently, HG is normal if and only if gHg−1 = H for any gG, i.e. if and only if each conjugacy class of G is either entirely inside H or entirely outside H. This is equivalent to saying that H is invariant under all inner automorphisms of G.

The property gH = Hg means that left and rights cosets of H in G coincide. From this one sees that the cosets form a group with the operation g1H * g2H = g1g2H which is called the factor group or quotient group of G by H, denoted by G/H.

In the special case that a subgroup H has only two cosets in G (namely H and gH for some g not contained in H), the subgroup H is always normal in G.

Connection with homomorphisms

If f is a homomorphism from G to another group, then the kernel of f is a normal subgroup of G. Conversely, every normal subgroup H [math]\triangleleft[/math] G arises as the kernel of a homomorphism, namely of the projection homomorphism GG/H defined by mapping g to its coset gH.

Example

The group T containing all the translations of a space group G is a normal subgroup in G called the translation subgroup of G. The factor group G/T is isomorphic to the point group P of G.