Theorem 5.14

Statement:
A subset $H$ of a group $G$ is a subgroup of $G$ if and only if
1. $H$ is closed under the binary operation of $G$,
2. the identity element $e$ of $G$ is in $H$,
3. for all $a\in H$ it is true that $a^{-1} \in H$ also.

Proof:
The fact that if $H \leq G$ then Conditions 1,2, and 3 must hold follows at once from the definition of a subgroup and from the remarks preceding Example 5.13 in the text.

Conversely, suppose $H$ is a subset of a group $G$ such that Conditions 1,2, and 3 hold. By 2 we have at once that $\mathscr{G_2}$ is satisfied. Also $\mathscr{G_3}$ is satisfied by 3. It remains to check the associative axiom, $\mathscr{G_1}$. But surely for all $a,b,c\in H$ it is true that $(ab)c=a(bc)$ in $H$, for we may actually view this as an equation in $G$, where the associative law holds. Hence $H\leq G$.