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Difference between revisions of "Normal subgroup"

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''normal divisor, invariant subgroup''
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{{MSC|20}}
  
A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n0676901.png" /> of a [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n0676902.png" /> for which the left decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n0676903.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n0676904.png" /> is the same as the right one; in other words, a subgroup such that for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n0676905.png" /> the cosets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n0676906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n0676907.png" /> are the same (as sets). In this case one also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n0676908.png" /> is normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n0676909.png" /> and writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769010.png" />; if also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769011.png" />, one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769012.png" />. A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769013.png" /> is normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769014.png" /> if and only if it contains all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769015.png" />-conjugates of any of its elements (see [[Conjugate elements|Conjugate elements]]), that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769016.png" />. A normal subgroup can also be defined as one that coincides with all its conjugates, as a consequence of which it is also known as a self-conjugate subgroup.
 
  
For any [[Homomorphism|homomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769017.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769018.png" /> of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769019.png" /> that are mapped to the unit element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769020.png" /> (the kernel of the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769021.png" />) is a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769022.png" />, and conversely, every normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769023.png" /> is the kernel of some homomorphism; in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769024.png" /> is the kernel of the canonical homomorphism onto the [[Quotient group|quotient group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769025.png" />.
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A ''normal subgrup'' (also: ''normal divisor, invariant subgroup'')
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is a subgroup $H$ of a
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[[Group|group]] $G$ for which the left decomposition of $G$ modulo $H$ is the same as the right one; in other words, a subgroup such that for any element $a\in G$ the cosets $aH$ and $Ha$ are the same (as sets). In this case one also says that $H$ is normal in $G$ and writes $H\trianglelefteq G$; if also $H\ne G$, one writes $H\triangleleft G$. A subgroup $H$ is normal in $G$ if and only if it contains all $G$-conjugates of any of its elements (see
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[[Conjugate elements|Conjugate elements]]), that is $H^G\subseteq H$. A normal subgroup can also be defined as one that coincides with all its conjugates, as a consequence of which it is also known as a self-conjugate subgroup.
  
The intersection of any set of normal subgroups is normal, and the subgroup generated by any system of normal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769026.png" /> is normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769027.png" />.
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For any
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[[Homomorphism|homomorphism]] $\varphi:F\to G^*$ the set $K$ of elements of $G$ that are mapped to the unit element of $G^*$ (the kernel of the homomorphism $\varphi$) is a normal subgroup of $G$, and conversely, every normal subgroup of $G$ is the kernel of some homomorphism; in particular, $K$ is the kernel of the canonical homomorphism onto the
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[[Quotient group|quotient group]] $G/K$.
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The intersection of any set of normal subgroups is normal, and the subgroup generated by any system of normal subgroups of $G$ is normal in $G$.
  
  
  
 
====Comments====
 
====Comments====
A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769028.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769029.png" /> is normal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769030.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769031.png" />, or, equivalently, if the normalizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769032.png" />, cf. [[Normalizer of a subset|Normalizer of a subset]]. A normal subgroup is also called an invariant subgroup because it is invariant under the inner automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769034.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067690/n06769035.png" />. A subgroup that is invariant under all automorphisms is called a fully-invariant subgroup or [[Characteristic subgroup|characteristic subgroup]]. A subgroup that is invariant under all endomorphisms is a [[Fully-characteristic subgroup|fully-characteristic subgroup]].
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A subgroup $H$ of a group $G$ is normal if $g^{-1}Hg = H$ for all $g\in G$, or, equivalently, if the normalizer $N_G(H) = G$, cf.
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[[Normalizer of a subset|Normalizer of a subset]]. A normal subgroup is also called an invariant subgroup because it is invariant under the inner automorphisms $x\mapsto x^g=g^{-1}xg$, $g\in G$, of $G$. A subgroup that is invariant under all automorphisms is called a fully-invariant subgroup or
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[[Characteristic subgroup|characteristic subgroup]]. A subgroup that is invariant under all endomorphisms is a
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[[Fully-characteristic subgroup|fully-characteristic subgroup]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Hall jr.,  "The theory of groups" , Macmillan (1959)  pp. 26</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience (1962)  pp. 5</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.G. Kurosh,  "The theory of groups" , '''1''' , Chelsea  (1955)  pp. Chapt. III  (Translated from Russian)</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|CuRe}}||valign="top"| C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras", Interscience (1962)  pp. 5  {{MR|0144979}}  {{ZBL|0131.25601}}
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|-
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|valign="top"|{{Ref|Ha}}||valign="top"| M. Hall jr.,  "The theory of groups", Macmillan (1959)  pp. 26  {{MR|0103215}}  {{ZBL|0084.02202}}
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|-
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|valign="top"|{{Ref|Ku}}||valign="top"| A.G. Kurosh,  "The theory of groups", '''1''', Chelsea  (1955)  pp. Chapt. III  (Translated from Russian) {{MR|0071422}} 
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|-
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|}

Revision as of 17:52, 1 March 2012

2020 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]


A normal subgrup (also: normal divisor, invariant subgroup) is a subgroup $H$ of a group $G$ for which the left decomposition of $G$ modulo $H$ is the same as the right one; in other words, a subgroup such that for any element $a\in G$ the cosets $aH$ and $Ha$ are the same (as sets). In this case one also says that $H$ is normal in $G$ and writes $H\trianglelefteq G$; if also $H\ne G$, one writes $H\triangleleft G$. A subgroup $H$ is normal in $G$ if and only if it contains all $G$-conjugates of any of its elements (see Conjugate elements), that is $H^G\subseteq H$. A normal subgroup can also be defined as one that coincides with all its conjugates, as a consequence of which it is also known as a self-conjugate subgroup.

For any homomorphism $\varphi:F\to G^*$ the set $K$ of elements of $G$ that are mapped to the unit element of $G^*$ (the kernel of the homomorphism $\varphi$) is a normal subgroup of $G$, and conversely, every normal subgroup of $G$ is the kernel of some homomorphism; in particular, $K$ is the kernel of the canonical homomorphism onto the quotient group $G/K$.

The intersection of any set of normal subgroups is normal, and the subgroup generated by any system of normal subgroups of $G$ is normal in $G$.


Comments

A subgroup $H$ of a group $G$ is normal if $g^{-1}Hg = H$ for all $g\in G$, or, equivalently, if the normalizer $N_G(H) = G$, cf. Normalizer of a subset. A normal subgroup is also called an invariant subgroup because it is invariant under the inner automorphisms $x\mapsto x^g=g^{-1}xg$, $g\in G$, of $G$. A subgroup that is invariant under all automorphisms is called a fully-invariant subgroup or characteristic subgroup. A subgroup that is invariant under all endomorphisms is a fully-characteristic subgroup.

References

[CuRe] C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras", Interscience (1962) pp. 5 MR0144979 Zbl 0131.25601
[Ha] M. Hall jr., "The theory of groups", Macmillan (1959) pp. 26 MR0103215 Zbl 0084.02202
[Ku] A.G. Kurosh, "The theory of groups", 1, Chelsea (1955) pp. Chapt. III (Translated from Russian) MR0071422
How to Cite This Entry:
Normal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_subgroup&oldid=21397
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article