Proper subgroup

Symbol-free definition
A subgroup of a group is said to be proper if it does not equal the whole group, or equivalently, if as a subset, it is a proper subset of the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed proper if $$H$$ is not the whole of $$G$$.

Opposite
The opposite of the property of being a proper subgroup is the property of being the improper subgroup, viz the whole group.

Related properties

 * Nontrivial subgroup is a subgroup that is not the trivial group

Metaproperties
Any subgroup of a proper subgroup is proper. That is, if $$H \le K \le G$$ and $$K$$ is proper in $$G$$, so is $$H$$.

Any proper subgroup of a subgroup is proper. That is, if $$H \le K \le G$$ and $$H$$ is proper in $$K$$, $$H$$ is also proper in $$G$$.