subgroup

A subset and superset $H\subseteq G$ is called a subgroup of $G$ if $H$ also forms a group under the same binary operation. This is often denoted $H\le G$.

The necessary and sufficient conditions for $H\subseteq G$ being a subgroup are:

1. $H$ is nonempty
2. every element of $H$ has an inverse in $H$
3. $H$ is closed under the operation

Formally,

1. $H\ne \varnothing$
2. $h\in H\Rightarrow h^{-1}\in H$
3. $\mathrm{\forall }{h}_1,h_2\in H,(h_1{h}_2)\in H$

Recall the definition of a group:

A group $G$ is a set with a binary operation such that the set is closed under the operation, the operation is associative, there is an identity element, and every element of the set has an inverse element.

Being closed fulfills the first criteria. Having the same binary operation as the group $G$ fulfills the second criteria of being a group. Having an inverse implicitly fulfills the third criteria, and explicitly fulfills the fourth criteria.

The trivial subgroup is $H=\{e\}$ (only the identity element)

A subgroup is proper (denoted $H<G$) if $H\ne G$