normal subgroup

A subgroup $H$ of a group $G$ is normal if $aH=Ha$ for all $a\\in G$ . Equivalently, $H\\subset G$ is normal if and only if $aHa^{-1}=H$ for all $a\\in G$ , i.e., if and only if each conjugacy class of $G$ is either entirely inside $H$ or entirely outside $H$ .

The notation $H\\trianglelefteq G$ or $H\\triangleleft G$ is often used to denote that $H$ is a normal subgroup of $G$ .

The kernel $\\ker(f)$ of any group homomorphism $f:G\\longrightarrow G^{\\prime}$ is a normal subgroup of $G$ . More surprisingly, the converse is also true: any normal subgroup $H\\subset G$ is the kernel of some homomorphism (one of these being the projection map $\\rho:G\\longrightarrow G/H$ , where $G/H$ is the quotient group).

单词	NormalSubgroup
释义	normal subgroup A subgroup $H$ of a group $G$ is normal if $aH=Ha$ for all $a\\in G$ . Equivalently, $H\\subset G$ is normal if and only if $aHa^{-1}=H$ for all $a\\in G$ , i.e., if and only if each conjugacy class of $G$ is either entirely inside $H$ or entirely outside $H$ . The notation $H\\trianglelefteq G$ or $H\\triangleleft G$ is often used to denote that $H$ is a normal subgroup of $G$ . The kernel $\\ker(f)$ of any group homomorphism $f:G\\longrightarrow G^{\\prime}$ is a normal subgroup of $G$ . More surprisingly, the converse is also true: any normal subgroup $H\\subset G$ is the kernel of some homomorphism (one of these being the projection map $\\rho:G\\longrightarrow G/H$ , where $G/H$ is the quotient group).
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