Proper normal subgroup

Symbol-free definition
A subgroup of a group is termed a proper normal subgroup if it satisfies both these conditions:


 * It is a proper subgroup i.e. it is not the whole group
 * It is a normal subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a proper normal subgroup if:


 * $$H \ne G$$ i.e. $$H$$ is not the whole of $$G$$
 * $$H \triangleleft G$$ i.e. $$H$$ is a normal subgroup of $$G$$

Stronger properties

 * Maximal normal subgroup

Incomparable properties

 * Nontrivial normal subgroup