Difference between revisions of "Normal subgroup"
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| − | <font color="blue"> | + | <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 ∈G. Equivalently, H ⊂ G is normal if and only if gHg<sup> | + | A [[subgroup]] ''H'' of a group ''G'' is '''normal''' in ''G'' (''H'' <math>\triangleleft</math> ''G'') if ''gH'' = ''Hg'' for any ''g'' ∈ ''G''. Equivalently, ''H'' ⊂ ''G'' is normal if and only if ''gHg''<sup>−1</sup> = ''H'' for any ''g'' ∈ ''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''. |
| − | gH = Hg means that left and rights [[coset]]s of H in G coincide. | + | The property ''gH'' = ''Hg'' means that left and rights [[coset]]s of ''H'' in ''G'' coincide. From this one sees that the cosets |
| + | 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 | ||
| + | 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 [[group homomorphism|homomorphism]] from ''G'' to another group, then the [[group homomorphism|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 ''G'' → ''G/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''. | ||
[[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 g ∈ G. Equivalently, H ⊂ G is normal if and only if gHg−1 = H for any g ∈ 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 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 G → G/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.