Normal subgroup/Kernel/Introduction/Section

Definition

Let ${}G$ be a group, and let ${}H\subseteq G$ denote a subgroup. ${}H$ is called a normal subgroup if

${}xH=Hx\,$

holds for all ${}x\in G$, that is, if every left coset

of ${}x$ coincides with the right coset of ${}x$.

For a normal subgroup, it is not necessary to distinguish between left cosets and right cosets; we just talk about cosets. Instead of ${}xH$ or ${}Hx$, we write usually ${}[x]$. The equality ${}xH=Hx$ does not mean that ${}xh=hx$ for all ${}h\in H$; it only means that for every ${}h\in H$, there exists a ${}{\tilde {h}}\in H$ fulfilling ${}xh={\tilde {h}}x$.

Lemma

Normal subgroup/Characterization/Fact

Proof

Normal subgroup/Characterization/Fact/Proof

$\Box$

Example

Permutation group S3/Subgroups and normal subgroup/Example

Lemma

Group homomorphism/Kernel/Normal subgroup/Fact

Proof

Group homomorphism/Kernel/Normal subgroup/Fact/Proof

$\Box$