Vipul: /* Notation and terminology */

2010-12-18T17:36:49Z

Notation and terminology

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	===Notation and terminology===		===Notation and terminology===

	For a subgroup <math>H \!</math> of a group <math>G \!</math>, we denote the normality of <math>\! H</math> in <math>\! G</math> by <math>H \underline{\triangleleft} G</math> or <math>G \underline{\triangleright} H</math>~~{{subgroup-notation-page}}~~. In words, we say that <math>\! H</math> is ''normal in'' <math>\! G</math> or a ''normal subgroup of'' <math>\! G</math>.		For a subgroup <math>H \!</math> of a group <math>G \!</math>, we denote the normality of <math>\! H</math> in <math>\! G</math> by <math>H \underline{\triangleleft} G</math> or <math>G \underline{\triangleright} H</math>. In words, we say that <math>\! H</math> is ''normal in'' <math>\! G</math> or a ''normal subgroup of'' <math>\! G</math>.

Vipul: Created page with '{{quotation|For a much more thorough and detailed treatment of normal subgroups, see normal subgroup at the group properties wiki}} ==Definition==...'

2010-12-18T17:35:28Z

Created page with '{{quotation|For a much more thorough and detailed treatment of normal subgroups, see normal subgroup at the group properties wiki}} ==Definition==...'

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{{quotation|For a much more thorough and detailed treatment of normal subgroups, see [[Groupprops:Normal subgroup|normal subgroup at the group properties wiki]]}}

==Definition==

{{quick phrase|[[quick phrase::invariant under inner automorphisms, self-conjugate subgroup]], [[quick phrase::same left and right cosets]], [[quick phrase::kernel of a homomorphism]], [[quick phrase::subgroup that is a union of conjugacy classes]]}}

===Definitions in tabular format===

Six equivalent definitions of normality are listed below. Note that each of these definitions (except the first one, as noted) assumes that we ''already'' have a group and a subgroup. Thus, to prove normality using any of these definitions, we first need to check that we actually have a subgroup.

{| class="sortable" border="1"
! No. !! Shorthand !! A [[subgroup]] of a [[group]] is normal in it if... !! A subgroup <math>H</math> of a group <math>G</math> is normal in <math>G</math> if ... !! Additional comments
|-
| 1 || Homomorphism kernel || it is the kernel of a [[Defining ingredient::homomorphism]] from the group. || there is a [[homomorphism]] <math>\varphi</math> from <math>G</math> to a group <math>K</math> such that the [[kernel]] of <math>\varphi</math> is precisely <math>H\!</math>. In other words, <math>\varphi(x) \!</math> is the identity element of <math>K</math> if and only if <math>x \in H</math>. || In this case, we do not need to separately check that <math>N</math> is a sub''group'' since the kernel of a homomorphism is automatically a subgroup.
|-
| 2 || Inner automorphism invariance || it is invariant under all [[Defining ingredient::inner automorphism]]s. || for all <math>g \in G</math>, <math>gHg^{-1} \subseteq H</math>. More explicitly, for all <math>g \in G, h \in H</math>, we have <math>ghg^{-1} \in H</math>. || Thus, normality is the [[invariance property]] with respect to the property of an automorphism being inner. This definition also motivates the term ''invariant subgroup'' for normal subgroup (which was used earlier).
|-
| 3 || Equals conjugates || it equals each of its [[Defining ingredient::conjugate subgroups|conjugates]] in the whole group. || for all <math>g</math> in <math>G</math>, <math>gHg^{-1} = H</math>. || This definition also motivates the term ''self-conjugate subgroup'' for normal subgroup (which was used earlier).
|-
| 4 || Left/right cosets equal || its [[Defining ingredient::left coset]]s are the same as its [[Defining ingredient::right coset]]s (that is, it commutes with every element of the group). || for all <math>g</math> in <math>G</math>, <math>gH = Hg</math>. || When we say <math>gH = Hg</math>, we only mean equality as sets. It is ''not'' necessary that <math>gh = hg</math> for <math>h \in H</math>. That stronger condition defines [[central subgroup]].
|-
| 5 || Union of conjugacy classes || it is a union of [[Defining ingredient::conjugacy class]]es. || <math>H</math> is a union of [[conjugacy class]]es in <math>G</math> ||
|-
| 6 || Commutator inside || it contains its [[commutator of two subgroups|commutator]] with the whole group. || the [[defining ingredient::commutator of two subgroups|commutator]] <math>[H,G]</math> (which coincides with the commutator <math>[G,H]</math>) is contained in <math>H</math>. ||
|}

===Notation and terminology===

For a subgroup <math>H \!</math> of a group <math>G \!</math>, we denote the normality of <math>\! H</math> in <math>\! G</math> by <math>H \underline{\triangleleft} G</math> or <math>G \underline{\triangleright} H</math>{{subgroup-notation-page}}. In words, we say that <math>\! H</math> is ''normal in'' <math>\! G</math> or a ''normal subgroup of'' <math>\! G</math>.

Normal subgroup - Revision history

Vipul: /* Notation and terminology */

Vipul: Created page with '{{quotation|For a much more thorough and detailed treatment of normal subgroups, see normal subgroup at the group properties wiki}} ==Definition==...'