Difference between revisions of "Normal subgroup"
From Online Dictionary of Crystallography
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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. | 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. | ||
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| + | ==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]] | ||
Revision as of 21:18, 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.
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.