Difference between revisions of "Normal subgroup"
From Online Dictionary of Crystallography
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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 ∈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 say that H is invariant under all [[automorphism|inner automorphisms]] of G. | 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 say 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== | ==Example== | ||
Revision as of 12:29, 2 April 2009
Sousgroupe normal (Fr); Sottogruppo normale (It); 正規部分群 (Ja)
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 say 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.