Normal subgroup

2010 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 $G$.

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.