Difference between revisions of "Normal subgroup"
From Online Dictionary of Crystallography
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== 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>-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. | ||
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| + | gH = Hg means that left and rights [[coset]]s of H in G coincide. As a consequence, every subgroup with only one other coset is normal. | ||
[[Category: Fundamental crystallography]] | [[Category: Fundamental crystallography]] | ||
Revision as of 18:03, 9 March 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.
gH = Hg means that left and rights cosets of H in G coincide. As a consequence, every subgroup with only one other coset is normal.