Difference between revisions of "Normal subgroup"
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| − | + | A ''normal subgrup'' (also: ''normal divisor, invariant subgroup'') | |
| + | is a subgroup $H$ of a | ||
| + | [[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 | ||
| + | [[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 | + | For any |
| + | [[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 | ||
| + | [[Quotient group|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==== | ====Comments==== | ||
| − | A subgroup | + | 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|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|characteristic subgroup]]. A subgroup that is invariant under all endomorphisms is a | ||
| + | [[Fully-characteristic subgroup|fully-characteristic subgroup]]. | ||
====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |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}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ha}}||valign="top"| M. Hall jr., "The theory of groups", Macmillan (1959) pp. 26 {{MR|0103215}} {{ZBL|0084.02202}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ku}}||valign="top"| A.G. Kurosh, "The theory of groups", '''1''', Chelsea (1955) pp. Chapt. III (Translated from Russian) {{MR|0071422}} | ||
| + | |- | ||
| + | |} | ||
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=18885
Normal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_subgroup&oldid=18885
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article