Automorphism - Revision history

BrianMcMahon: Tidied translations.

2017-11-08T18:11:38Z

Tidied translations.

BrianMcMahon: Style edits to align with printed edition

2017-05-12T12:45:52Z

Style edits to align with printed edition

BerndSouvignier at 10:40, 2 April 2009

2009-04-02T10:40:43Z

BerndSouvignier: adjusted inner automorphisms, some further small changes

2009-04-02T10:39:26Z

adjusted inner automorphisms, some further small changes

MassimoNespolo: Internal links

2009-03-19T14:40:55Z

Internal links

MassimoNespolo: link

2009-03-18T18:39:35Z

link

MassimoNespolo at 14:52, 24 April 2007

2007-04-24T14:52:01Z

MassimoNespolo at 14:22, 23 April 2007

2007-04-23T14:22:01Z

MassimoNespolo at 14:20, 23 April 2007

2007-04-23T14:20:14Z

MassimoNespolo at 14:05, 23 April 2007

2007-04-23T14:05:58Z

← Older revision		Revision as of 18:11, 8 November 2017
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−	<~~Font~~ color="blue">Automorphisme</font> (''Fr''). <~~Font~~ color="red">Automorphismus</font> (''Ge''). <~~Font~~ color="~~green~~">Automorfismo</font> (''Sp''). <~~Font~~ color="~~black~~">~~Automorfismo~~</font> (''It''). <~~Font~~ color="~~purple~~">~~自己同形~~</font> (''Ja'').	+	<font color="blue">Automorphisme</font> (''Fr''). <font color="red">Automorphismus</font> (''Ge''). <font color="black">Automorfismo</font> (''It''). <font color="purple">自己同形</font> (''Ja''). <font color="green">Automorfismo</font> (''Sp'').

	==Definition==		==Definition==

@@ Line 2: / Line 2: @@
 ==Definition==
-An [[Group isomorphism| isomorphism]] from a group (''G'',*) to itself is called an '''automorphism''' of this group. It is a [[mapping|bijection]] ''f'' : ''G'' → ''G'' such that
+An [[Group isomorphism|isomorphism]] from a group (''G'',*) to itself is called an '''automorphism''' of this group. It is a [[mapping|bijection]] ''f'' : ''G'' → ''G'' such that
 <div align="center">
@@ Line 8: / Line 8: @@
 </div>
-An automorphism preserves the structural properties of a group, e.g.:
+An automorphism preserves the structural properties of a group, ''e.g.''
 * The identity element of ''G'' is mapped to itself.
 * [[Subgroup]]s are mapped to subgroups, [[normal subgroup]]s to normal subgroups.
@@ Line 35: / Line 35: @@
 The inner automorphisms form a [[normal subgroup]] of '''Aut(''G'')''', called the '''inner automorphism group''' and denoted by '''Inn(''G'')'''.
-The inner automorphism group is [[group isomorphism|isomorphic]] to the [[factor group|quotient]] of ''G'' by its [[center]] ''Z(G)''. In particular, for [[Abelian group]]s the inner automorphism group consists just of the trivial automorphism.
+The inner automorphism group is [[group isomorphism|isomorphic]] to the [[factor group|quotient]] of ''G'' by its [[center|centre]] ''Z(G)''. In particular, for [[Abelian group]]s the inner automorphism group consists just of the trivial automorphism.
 ==Outer automorphism==

@@ Line 10: / Line 10: @@
 An automorphism preserves the structural properties of a group, e.g.:
 * The identity element of ''G'' is mapped to itself.
-* [[Subgroups]] are mapped to subgroups, [[normal subgroup]]s to normal subgroups.
+* [[Subgroup]]s are mapped to subgroups, [[normal subgroup]]s to normal subgroups.
 * [[Conjugacy class]]es are mapped to conjugacy classes (the same or another).
 * The [[image]] ''f(g)'' of an element ''g'' has the same [[order]] as ''g''.

@@ Line 5: / Line 5: @@
 <div align="center">
-''f'' (''u'') * ''f'' (''v'') = ''f'' (''u'' * ''v'')
+''f'' (''g'') * ''f'' (''h'') = ''f'' (''g'' * ''h'')
 </div>
-An automorphism always maps the identity to itself. The image under an automorphism of a [[conjugacy class]] is always a conjugacy class (the same or another). The [[image]] of an element has the same order as that element.
+An automorphism preserves the structural properties of a group, e.g.:
-The composition of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group ''G'', denoted by '''Aut(''G'')''', forms itself a group, the ''automorphism group'' of ''G''.
+The composition of two automorphisms is again an automorphism, and with composition as binary operation the set of all automorphisms of a group ''G'', denoted by '''Aut(''G'')''', forms itself a group, the ''automorphism group'' of ''G''.
 ==Inner automorphism==
@@ Line 16: / Line 20: @@
 <div align="center">
-''f'' : ''G'' &rarr; ''G''
+''f<sub>a</sub>'' : ''G'' &rarr; ''G''
 </div>
@@ Line 22: / Line 26: @@
 <div align="center">
-''f''(''x'') = ''axa''<sup>−1</sup>
+''f<sub>a</sub>(g)'' = ''aga<sup>−1</sup>''
 </div>
-where ''a'' is a given fixed element of ''G'', for all ''x'' in ''G''.
+for all ''g'' in ''G'', where ''a'' is a given fixed element of ''G''.
-The operation ''axa''<sup>−1</sup> is called '''conjugation''' (see also [[conjugacy class]]).
+The operation ''aga<sup>−1</sup>'' is called '''conjugation''' by ''a'' (see also [[conjugacy class]]).
 ==Outer automorphism==
-The '''outer automorphism group''' of a group ''G'' is the [[factor group|quotient]] of the automorphism group '''Aut(''G'')''' by its inner automorphism group '''Inn(''G'')'''. The outer automorphism group is usually denoted '''Out(''G'')'''.
+The '''outer automorphism group''' of a group ''G'' is the [[factor group|quotient]] of the automorphism group '''Aut(''G'')''' by its inner automorphism group '''Inn(''G'')'''. The outer automorphism group is usually denoted by '''Out(''G'')'''.
-For all Abelian groups there is at least the automorphism that replaces the group elements by their inverses. All other automorphisms are outer automorphisms. Non-Abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms.
+For Abelian groups the mapping ''g'' &rarr; ''g<sup>-1</sup>'' is an outer automorphism, whereas for non-Abelian groups this mapping is not even a [[group homomorphism|homomorphism]].
 ==See also==

← Older revision		Revision as of 14:40, 19 March 2009
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	For all Abelian groups there is at least the automorphism that replaces the group elements by their inverses. All other automorphisms are outer automorphisms. Non-Abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms.		For all Abelian groups there is at least the automorphism that replaces the group elements by their inverses. All other automorphisms are outer automorphisms. Non-Abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms.

		+	==See also==
		+	*[[Mapping]]
		+	*[[Group homomorphism]]
		+	*[[Group isomorphism]]

	[[Category:Fundamental crystallography]]		[[Category:Fundamental crystallography]]

← Older revision		Revision as of 14:52, 24 April 2007
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−	<Font color="blue">Automorphisme</font> (''Fr''). <Font color="black">Automorfismo</font> (''It''). <Font color="purple">自己同形</font> (''Ja'').	+	<Font color="blue">Automorphisme</font> (''Fr''). <Font color="red">Automorphismus</font> (''Ge''). <Font color="green">Automorfismo</font> (''Sp''). <Font color="black">Automorfismo</font> (''It''). <Font color="purple">自己同形</font> (''Ja'').

	==Definition==		==Definition==

← Older revision		Revision as of 14:22, 23 April 2007
Line 1:		Line 1:
−	<Font color="blue">Automorphisme</font> (''Fr''). <Font color="black">Automorfismo</font> (''It'').　<Font color="purple">自己同形</font> (''Ja'').	+	<Font color="blue">Automorphisme</font> (''Fr''). <Font color="black">Automorfismo</font> (''It''). <Font color="purple">自己同形</font> (''Ja'').

	==Definition==		==Definition==

← Older revision		Revision as of 14:20, 23 April 2007
Line 1:		Line 1:
−	<Font color="blue">Automorphisme</font> (''Fr''). <Font color="black">Automorfismo</font> (''It'').	+	<Font color="blue">Automorphisme</font> (''Fr''). <Font color="black">Automorfismo</font> (''It'').　<Font color="purple">自己同形</font> (''Ja'').

	==Definition==		==Definition==