View source for 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.

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====
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====
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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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|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}}  
|-
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