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Difference between revisions of "Normal subgroup"

From Online Dictionary of Crystallography

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<font color="blue"> Sousgroupe 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 ==
A [[subgroup]] H of a group G is '''normal''' in G (H <math>\triangleleft</math> G) if gH = Hg for any g &isin;G. Equivalently, H &sub; G is normal if and only if gHg<sup>-1</sup> = H for any g &isin;G, i.e., if and only if each [[conjugacy class]] of G is either entirely inside H or entirely outside H. This is equivalent to say that H is invariant under all [[automorphism|inner automorphisms]] of G.
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A [[subgroup]] ''H'' of a group ''G'' is '''normal''' in ''G'' (''H'' <math>\triangleleft</math> ''G'') if ''gH'' = ''Hg'' for any ''g'' &isin; ''G''. Equivalently, ''H'' &sub; ''G'' is normal if and only if ''gHg''<sup>&minus;1</sup> = ''H'' for any ''g'' &isin; ''G'', ''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 [[automorphism|inner automorphisms]] of ''G''.
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The property ''gH'' = ''Hg'' means that left and rights [[coset]]s of ''H'' in ''G'' coincide. From this one sees that the cosets
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form a group with the operation ''g''<sub>1</sub>''H'' * ''g''<sub>2</sub>''H'' = ''g''<sub>1</sub>''g''<sub>2</sub>''H'' which is called
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the [[factor group]] or '''quotient group''' of ''G'' by ''H'', denoted by ''G/H''.
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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''.
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==Connection with homomorphisms==
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If ''f'' is a [[group homomorphism|homomorphism]] from ''G'' to another group, then the [[group homomorphism|kernel]]
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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 ''G'' &rarr; ''G/H'' defined by mapping ''g'' to its [[coset]] ''gH''.
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==Example==
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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''.
  
 
[[Category: Fundamental crystallography]]
 
[[Category: Fundamental crystallography]]

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.