Normal subgroup

Definitions in tabular format
Six equivalent definitions of normality are listed below. Note that each of these definitions (except the first one, as noted) assumes that we already have a group and a subgroup. Thus, to prove normality using any of these definitions, we first need to check that we actually have a subgroup.

Notation and terminology
For a subgroup $$H \!$$ of a group $$G \!$$, we denote the normality of $$\! H$$ in $$\! G$$ by $$H \underline{\triangleleft} G$$ or $$G \underline{\triangleright} H$$. In words, we say that $$\! H$$ is normal in $$\! G$$ or a normal subgroup of $$\! G$$.