Improper subgroup

Symbol-free definition
A subgroup of a group is termed improper if it equals the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed improper if $$H = G$$.

Opposite
The negation of the subgroup property of being improper is the subgroup property of being proper.

Metaproperties
The property of being improper is transitive: the improper subgroup of the improper subgroup is improper. In fact, it is a.