Group homomorphism - Revision history

BrianMcMahon: Tidied translations.

2017-12-01T09:19:36Z

Tidied translations.

BrianMcMahon: Tidied translations.

2017-11-13T14:44:50Z

Tidied translations.

BrianMcMahon: Style edits to align with printed edition

2017-05-15T12:30:18Z

Style edits to align with printed edition

BrianMcMahon at 13:30, 6 February 2012

2012-02-06T13:30:49Z

BerndSouvignier: definition of a group replaced by a link to a separate definition

2009-04-01T14:46:09Z

definition of a group replaced by a link to a separate definition

BerndSouvignier: some small adjustments

2009-04-01T14:05:21Z

some small adjustments

MassimoNespolo: /* Kernel and image */ Bold

2009-03-19T14:36:48Z

‎Kernel and image: Bold

MassimoNespolo: /* Types of homomorphisms */ More details and figures

2009-03-19T14:35:59Z

‎Types of homomorphisms: More details and figures

MassimoNespolo: /* Homorphisms between G and H ≠ G */

2009-03-19T14:14:40Z

‎Homorphisms between G and H ≠ G

MassimoNespolo: /* Homorphisms between G and H ≠ G */

2009-03-19T14:14:18Z

‎Homorphisms between G and H ≠ G

← Older revision		Revision as of 09:19, 1 December 2017
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−	<font color="blue">Homomorphisme de groupes</font> (''Fr''). <font color="red">Gruppenhomomorphismus</font> (''Ge''). <font color="black">Omomorfismo di gruppi </font> (''It''). <~~Font~~ color="purple">準同形</font> (''Ja''). <font color="green">Homomorfismo de grupos </font> (''Sp'').	+	<font color="blue">Homomorphisme de groupes</font> (''Fr''). <font color="red">Gruppenhomomorphismus</font> (''Ge''). <font color="black">Omomorfismo di gruppi</font> (''It''). <font color="purple">準同形</font> (''Ja''). <font color="green">Homomorfismo de grupos</font> (''Sp'').

← Older revision		Revision as of 14:44, 13 November 2017
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−	<font color="blue">Homomorphisme de groupes </font> (''Fr''). <font color="red">Gruppenhomomorphismus</font> (''Ge''). <font color="~~green~~">~~Homomorfismo de grupos~~ </font> (''Sp''). <~~font~~ color="~~black~~">~~Omomorfismo di gruppi~~ </font> (''It''). <~~Font~~ color="~~purple~~">~~準同形~~</font> (''Ja'').	+	<font color="blue">Homomorphisme de groupes</font> (''Fr''). <font color="red">Gruppenhomomorphismus</font> (''Ge''). <font color="black">Omomorfismo di gruppi </font> (''It''). <Font color="purple">準同形</font> (''Ja''). <font color="green">Homomorfismo de grupos </font> (''Sp'').

@@ Line 5: / Line 5: @@
 ==Homomorphism between groups==
-A '''group homomorphism''' from a [[group]] (''G'', *) to a [[group]] (''H'', #) is a [[mapping]] ''f'' : ''G'' &rarr; ''H'' that preserves the composition law, i.e. for all ''u'' and ''v'' in ''G'' one has:
+A '''group homomorphism''' from a [[group]] (''G'', *) to a [[group]] (''H'', #) is a [[mapping]] ''f'' : ''G'' &rarr; ''H'' that preserves the composition law, ''i.e.'' for all ''u'' and ''v'' in ''G'' one has:
 <div align="center">
 ''f''(''u'' * ''v'') = ''f''(''u'') # ''f''(''v'').
 </div>
+A homomorphism ''f'' maps the identity element 1<sub>''G''</sub> of ''G'' to the identity element 1<sub>''H''</sub> of ''H'', and it also maps inverses to inverses: ''f''(''u''<sup>&minus;1</sup>) = ''f''(''u'')<sup>&minus;1</sup>.
 == Kernel and image==
-[[Image:Homomorphism.jpg|thumb|240px|'''Group homomorphism'''. The kernel of the homomorphism, ker( ''f'' ) is the set of elements of ''G'' that are mapped to the identity element of ''H''. The image of the homomorphism, im( ''f'' ), is the set of elements of ''H'' to which at least one element of ''G'' is mapped. im( ''f'' ) is not required to be the whole of ''H'']]
+[[Image:Homomorphism.jpg|thumb|240px|'''Group homomorphism'''. The kernel of the homomorphism, ker(''f''), is the set of elements of ''G'' that are mapped to the identity element of ''H''. The image of the homomorphism, im(''f''), is the set of elements of ''H'' to which at least one element of ''G'' is mapped. im(''f'') is not required to be the whole of ''H''.]]
 The ''kernel'' of the homomorphism ''f'' is the set of elements of ''G'' that are [[mapping|mapped]] to the identity of ''H'':
 <div align="center">
-ker( ''f'' ) = { ''u'' in ''G'' : ''f''(''u'') = 1<sub>''H''</sub> }
+ker(''f'') = { ''u'' in ''G'' : ''f''(''u'') = 1<sub>''H''</sub> }.
 </div>
 The ''image'' of the homomorphism ''f'' is the subset of elements of ''H'' to which at least one element of ''G'' is mapped by ''f'':
 <div align="center">
-im( ''f'' ) = { ''f''(''u'') : ''u'' in ''G'' }.
+im(''f'') = { ''f''(''u'') : ''u'' in ''G'' }.
 </div>
@@ Line 33: / Line 32: @@
 == Types of homomorphisms ==
-Homomorphisms can be classified according to different criteria, among which the relation between ''G'' and ''H'' and the nature of the mapping.
+Homomorphisms can be classified according to different criteria, among which are the relation between ''G'' and ''H'' and the nature of the mapping.
 ===Surjective, injective and bijective homomorphisms===
@@ Line 39: / Line 38: @@
 [[Image:Epimorphism.jpg|thumb|240px|An '''epimorphism''' is a surjective homomorphism, i.e. a homomorphism such that the image is the entire [[Mapping|codomain]]]]
-An '''epimorphism''' is a [[mapping|surjective]] homomorphism, that is, a homomorphism which is ''onto'' as a mapping. The image of the homomorphism is the whole of ''H'', i.e. im( ''f'' ) = ''H''
+An '''epimorphism''' is a [[mapping|surjective]] homomorphism, that is, a homomorphism which is ''onto'' as a mapping. The image of the homomorphism is the whole of ''H'', ''i.e.'' im(''f'') = ''H''.
-[[Image:Monomorphism.jpg|thumb|240px|A '''monomorphism''' is an injective homomorphism, i.e. a homomorphism where different elements of ''G'' are mapped to different elements of ''H'']]
+[[Image:Monomorphism.jpg|thumb|240px|A '''monomorphism''' is an injective homomorphism, ''i.e.'' a homomorphism where different elements of ''G'' are mapped to different elements of ''H''.]]
 A '''monomorphism''' is an [[mapping|injective]] homomorphism, that is, a homomorphism which is ''one-to-one'' as a mapping. In this case, ker( ''f'' ) = {1<sub>''G''</sub> }.
-[[Image:Isomorphism.jpg|thumb|240px|An '''isomorphism''' is a bijective homomorphism, i.e. it is a one-to-one correspondence between the elements of ''G'' and those of ''H''. Isomorphic groups (''G'',*) and (''H'',#) differ only in the notation of their elements and binary operations]]
+[[Image:Isomorphism.jpg|thumb|240px|An '''isomorphism''' is a bijective homomorphism, ''i.e.'' it is a one-to-one correspondence between the elements of ''G'' and those of ''H''. Isomorphic groups (''G'',*) and (''H'',#) differ only in the notation of their elements and binary operations.]]
 If the homomorphism ''f'' is a [[mapping|bijection]], then its inverse is also a group homomorphism, and ''f'' is called an '''[[group isomorphism| isomorphism]]'''; the groups (''G'',*) and (''H'',#) are called ''isomorphic'' and differ only in the notation of their elements (and possibly their binary operations), while they can be regarded as identical for most practical purposes.
@@ Line 53: / Line 52: @@
 ===Homomorphisms from a group to itself (G = H)===
-[[Image:Endomorphism.jpg|thumb|240px|An '''endomorphism''' is homomorphism of ''G'' to itself]]
+[[Image:Endomorphism.jpg|thumb|240px|An '''endomorphism''' is a homomorphism of ''G'' to itself.]]
-An '''endomorphism''' is a homomorphism of a group to itself: ''f'': ''G'' &rarr; ''G''.
+An '''endomorphism''' is a homomorphism of a group to itself: ''f'' : ''G'' &rarr; ''G''.
 A [[mapping|bijective]] (invertible) endomorphism (which is hence an [[group isomorphism| isomorphism]]) is called an '''[[automorphism]]'''. The kernel of the automorphism is the identity of ''G'' (1<sub>G</sub>) and the image of the automorphism coincides with ''G''. The set of all automorphisms of a group (''G'',*) forms itself a group, the ''automorphism group'' of ''G'', '''Aut(''G'')'''.
 [[Category:Fundamental crystallography]]

@@ Line 1: / Line 1: @@
 <font color="blue">Homomorphisme de groupes </font> (''Fr''). <font color="red">Gruppenhomomorphismus</font> (''Ge''). <font color="green">Homomorfismo de grupos </font> (''Sp''). <font color="black">Omomorfismo di gruppi </font> (''It''). <Font color="purple">準同形</font> (''Ja'').
 ==Homomorphism between groups==
 A '''group homomorphism''' from a [[group]] (''G'', *) to a [[group]] (''H'', #) is a [[mapping]] ''f'' : ''G'' &rarr; ''H'' that preserves the composition law, i.e. for all ''u'' and ''v'' in ''G'' one has:
 <div align="center">
@@ Line 10: / Line 12: @@
 A homomorphism ''f'' maps the identity element 1<sub>''G''</sub> of ''G'' to the identity element 1<sub>''H''</sub> of ''H'', and it also maps inverses to inverses: ''f''(''u''<sup>-1</sup>) = ''f''(''u'')<sup>-1</sup>.
 == Kernel and image==
-[[Image:Homomorphism.jpg|thumb|300px|'''Group homomorphism'''. The kernel of the homomorphism, ker( ''f'' ) is the set of elements of ''G'' that are mapped to the identity element of ''H''. The image of the homomorphism, im( ''f'' ), is the set of elements of ''H'' to which at least one element of ''G'' is mapped. im( ''f'' ) is not required to be the whole of ''H'']].
+[[Image:Homomorphism.jpg|thumb|240px|'''Group homomorphism'''. The kernel of the homomorphism, ker( ''f'' ) is the set of elements of ''G'' that are mapped to the identity element of ''H''. The image of the homomorphism, im( ''f'' ), is the set of elements of ''H'' to which at least one element of ''G'' is mapped. im( ''f'' ) is not required to be the whole of ''H'']]
 The ''kernel'' of the homomorphism ''f'' is the set of elements of ''G'' that are [[mapping|mapped]] to the identity of ''H'':
 <div align="center">
@@ Line 25: / Line 29: @@
 The kernel is a [[normal subgroup]] of ''G'' and the image is a [[subgroup]] of ''H''.
 == Types of homomorphisms ==
@@ Line 32: / Line 37: @@
 ===Surjective, injective and bijective homomorphisms===
-[[Image:Epimorphism.jpg|thumb|300px|An '''epimorphism''' is a surjective homomorphism, i.e. a homomorphism such that the image is the entire [[Mapping|codomain]]]].
+[[Image:Epimorphism.jpg|thumb|240px|An '''epimorphism''' is a surjective homomorphism, i.e. a homomorphism such that the image is the entire [[Mapping|codomain]]]]
 An '''epimorphism''' is a [[mapping|surjective]] homomorphism, that is, a homomorphism which is ''onto'' as a mapping. The image of the homomorphism is the whole of ''H'', i.e. im( ''f'' ) = ''H''
-[[Image:Monomorphism.jpg|thumb|300px|A '''monomorphism''' is an injective homomorphism, i.e. a homomorphism where different elements of ''G'' are mapped to different elements of ''H'']].
+[[Image:Monomorphism.jpg|thumb|240px|A '''monomorphism''' is an injective homomorphism, i.e. a homomorphism where different elements of ''G'' are mapped to different elements of ''H'']]
 A '''monomorphism''' is an [[mapping|injective]] homomorphism, that is, a homomorphism which is ''one-to-one'' as a mapping. In this case, ker( ''f'' ) = {1<sub>''G''</sub> }.
-[[Image:Isomorphism.jpg|thumb|300px|An '''isomorphism''' is a bijective homomorphism, i.e. it is a one-to-one correspondence between the elements of ''G'' and those of ''H''. Isomorphic groups (''G'',*) and (''H'',#) differ only in the notation of their elements and binary operations]].
+[[Image:Isomorphism.jpg|thumb|240px|An '''isomorphism''' is a bijective homomorphism, i.e. it is a one-to-one correspondence between the elements of ''G'' and those of ''H''. Isomorphic groups (''G'',*) and (''H'',#) differ only in the notation of their elements and binary operations]]
 If the homomorphism ''f'' is a [[mapping|bijection]], then its inverse is also a group homomorphism, and ''f'' is called an '''[[group isomorphism| isomorphism]]'''; the groups (''G'',*) and (''H'',#) are called ''isomorphic'' and differ only in the notation of their elements (and possibly their binary operations), while they can be regarded as identical for most practical purposes.
 ===Homomorphisms from a group to itself (G = H)===
-[[Image:Endomorphism.jpg|thumb|300px|An '''endomorphism''' is homomorphism of ''G'' to itself]].
+[[Image:Endomorphism.jpg|thumb|240px|An '''endomorphism''' is homomorphism of ''G'' to itself]]
 An '''endomorphism''' is a homomorphism of a group to itself: ''f'': ''G'' &rarr; ''G''.
 A [[mapping|bijective]] (invertible) endomorphism (which is hence an [[group isomorphism| isomorphism]]) is called an '''[[automorphism]]'''. The kernel of the automorphism is the identity of ''G'' (1<sub>G</sub>) and the image of the automorphism coincides with ''G''. The set of all automorphisms of a group (''G'',*) forms itself a group, the ''automorphism group'' of ''G'', '''Aut(''G'')'''.
 [[Category:Fundamental crystallography]]

@@ Line 1: / Line 1: @@
 <font color="blue">Homomorphisme de groupes </font> (''Fr''). <font color="red">Gruppenhomomorphismus</font> (''Ge''). <font color="green">Homomorfismo de grupos </font> (''Sp''). <font color="black">Omomorfismo di gruppi </font> (''It''). <Font color="purple">準同形</font> (''Ja'').
 ==Homomorphism between groups==
-A '''group homomorphism''' from (''G'', *) to (''H'', #) is a [[mapping]] ''f'' : ''G'' &rarr; ''H'' that preserves the composition law, ''i''.''e''. for all ''u'' and ''v'' in ''G'' one has:
+A '''group homomorphism''' from a [[group]] (''G'', *) to a [[group]] (''H'', #) is a [[mapping]] ''f'' : ''G'' &rarr; ''H'' that preserves the composition law, i.e. for all ''u'' and ''v'' in ''G'' one has:
 <div align="center">
 ''f''(''u'' * ''v'') = ''f''(''u'') # ''f''(''v'').

@@ Line 15: / Line 15: @@
 == Kernel and image==
-[[Image:Homomorphism.jpg|thumb|300px|Group homomorphism. The kernel of the homomorphism, ker( ''f'' ) is the set of elements of ''G'' that are mapped on the identity element of ''H''. The image of the homomorphism, im( ''f'' ), is the set of elements of ''H'' with one or more corresponding element in ''G''. im( ''f'' ) is not required to span the whole ''H'']].
+[[Image:Homomorphism.jpg|thumb|300px|'''Group homomorphism'''. The kernel of the homomorphism, ker( ''f'' ) is the set of elements of ''G'' that are mapped on the identity element of ''H''. The image of the homomorphism, im( ''f'' ), is the set of elements of ''H'' with one or more corresponding element in ''G''. im( ''f'' ) is not required to span the whole ''H'']].
 The ''kernel'' of the homomorphism is the set of the elements of ''G'' that are [[mapping|mapped]] on the identity of ''H'':
 <div align="center">

@@ Line 34: / Line 34: @@
 ===Homomorphisms mapping elements of different sets (G &ne; H)===
-[[Image:Epimorphism.jpg|thumb|300px|An epimorphism is a surjective homomorphism, ''i''.''e''. a homomorphism where the image spans the whole [[Mapping|codomain]]]]
+[[Image:Epimorphism.jpg|thumb|300px|An '''epimorphism''' is a surjective homomorphism, ''i''.''e''. a homomorphism where the image spans the whole [[Mapping|codomain]]]].
 An '''epimorphism''' is a [[mapping|surjective]] homomorphism, that is, a homomorphism which is ''onto'' as a function. The image of the homomorphism spans the whole set ''H'': in this case, img( ''f'' ) = ''H''
 A '''monomorphism''' is an [[mapping|injective]] homomorphism, that is, a homomorphism which is ''one-to-one'' as a function. In this case, ker( ''f'' ) = {1<sub>''G''</sub> }.
 If the homomorphism ''f'' is a [[mapping|bijection]], then its inverse is also a group homomorphism, and ''f'' is called an '''[[group isomorphism| isomorphism]]'''; the groups (''G'',*) and (''H'',#) are called ''isomorphic'' and differ only in the notation of their elements, while they are identical for all practical purposes.
 ===Homomorphisms mapping elements of the same set (G = H)===
 An '''endomorphism''' is a homomorphism of a group onto itself: ''f'': ''G'' &rarr; ''G''.
 A [[mapping|bijective]] (invertible) endomorphism (which is hence an [[group isomorphism| isomorphism]]) is called an '''[[automorphism]]'''. The set of all automorphisms of a group ''G'' forms itself a group, the ''automorphism group'' of ''G'', '''Aut(''G'')'''.
 [[Category:Fundamental crystallography]]