Definition 1. Let $(G, *)$ be a group. $(H, *)$ is a subgroup of $(G, *)$, written $H \leq G$, if $(H, *)$ itself is a group and $H \subseteq G$.
Proposition 1. All subgroups are neutral-preserving: If $G$ is a group with neutral element $e$ and $H$ is a subgroup, then the neutral element of $H$ is also $e$.
Proof. If $e’$ is the neutral element of $H$,
\[e'e' = e' = ee',\]hence $e’ = e$ by cancellation property of group.
Proposition 2 (Subgroup Test A). Let $G$ be a group and $H \subseteq G$ be non-empty. $H$ is a subgroup as soon as:
-
(Closure under multiplication) For all $a$, $b \in H$,
\[ab \in H.\] -
(Closure under inversion) For all $a \in H$,
\[a^{-1} \in H.\]
Proposition 3 (Subgroup Test B). Let $G$ be a group and $H \subseteq G$ be non-empty. $H$ is a subgroup as soon as for all $a$, $b\in H$,
\[ab^{-1} \in H.\]Proposition 4 (Subgroup Test C). Let $G$ be a finite group and $H\subseteq G$ be non-empty. $H$ is a subgroup as soon as $H$ is closed under multiplication, i.e for all $a$, $b\in H$,
\[ab \in H.\]