Normal subgroup - Revision history

BrianMcMahon: Added German and Spanish translations (U. Mueller)

2017-11-16T13:14:10Z

Added German and Spanish translations (U. Mueller)

BrianMcMahon at 10:28, 8 July 2017

2017-07-08T10:28:57Z

BrianMcMahon: Style edits to align with printed edition

2017-05-16T10:06:29Z

Style edits to align with printed edition

BerndSouvignier: added connection with homomorphisms

2009-04-02T12:29:49Z

added connection with homomorphisms

MassimoNespolo at 21:18, 9 March 2009

2009-03-09T21:18:58Z

MassimoNespolo at 18:03, 9 March 2009

2009-03-09T18:03:01Z

MassimoNespolo at 18:00, 9 March 2009

2009-03-09T18:00:43Z

MassimoNespolo at 16:43, 27 February 2007

2007-02-27T16:43:04Z

MassimoNespolo at 15:48, 27 February 2007

2007-02-27T15:48:30Z

MassimoNespolo at 19:40, 24 February 2007

2007-02-24T19:40:38Z

New page

Sousgroupe normal (''Fr''); Sottogruppo normale (''It'').

== Definition ==
A subgroup H of a group G is '''normal''' 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.

[[Category: Fundamental crystallography]]

← Older revision		Revision as of 13:14, 16 November 2017
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−	<font color="blue"> Sous-groupe normal </font> (''Fr''). <font color="black"> Sottogruppo normale </font> (''It''). <font color="purple"> 正規部分群 </font> (''Ja'').	+	<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 ==

← Older revision		Revision as of 10:28, 8 July 2017
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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'').	+	<font color="blue"> Sous-groupe normal </font> (''Fr''). <font color="black"> Sottogruppo normale </font> (''It''). <font color="purple"> 正規部分群 </font> (''Ja'').

	== Definition ==		== Definition ==

@@ Line 1: / Line 1: @@
-<font color="blue"> Sousgroupe normal </font> (''Fr''); <font color="black"> Sottogruppo normale </font> (''It''); <font color="purple"> 正規部分群 </font> (''Ja'')
+<font color="blue"> Sousgroupe normal </font> (''Fr''). <font color="black"> Sottogruppo normale </font> (''It''). <font color="purple"> 正規部分群 </font> (''Ja'').
 == 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.
+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''.
-The property gH = Hg means that left and rights [[coset]]s of H in G coincide. From this one sees that the cosets
+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
+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''.
@@ Line 13: / Line 13: @@
 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,
+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''.
 ==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.
+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]]

@@ Line 4: / Line 4: @@
 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.
-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.
+The property gH = Hg means that left and rights [[coset]]s of H in G coincide. From this one sees that the cosets
 ==Example==

← Older revision		Revision as of 21:18, 9 March 2009
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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.
		+
		+	==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]]

@@ Line 2: / Line 2: @@
 == Definition ==
-A [[subgroup]] H of a group G is '''normal''' 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.
+A [[subgroup]] H of a group G is '''normal''' (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.
 [[Category: Fundamental crystallography]]