Sufficiency of subgroup criterion

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This article gives a proof/explanation of the equivalence of multiple definitions for the term subgroup
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Statement

For a subset $H$ of a group $G$, the following are equivalent:

1. $H$ is a subgroup, viz $H$ is closed under the binary operation of multiplication, the inverse map, and contains the identity element
2. $H$ is a nonempty set closed under left quotient of elements (that is, for any $a,b$ in $H$, $b^{-1}a$ is also in $H$)
3. $H$ is a nonempty set closed under right quotient of elements (that is, for any $a,b$ in $H$, $ab^{-1}$ is also in $H$)

Proof

We shall here prove the equivalence of the first two conditions. Equivalence of the first and third conditions follows by analogous reasoning.

(1) implies (2)

Clearly, if $H$ is a subgroup:

• $H$ is nonempty since $H$ contains the identity element
• Whenever $a,b$ are in $H$ so is $b^{-1}$ and hence $b^{-1}a$

(2) implies (1)

Suppose $H$ is a nonempty subset closed under left quotient of elements. Then, pick an element $u$ from $H$.

• $e$ is in $H$: St $a=b=u$ to get $u^{-1}u$ is contained in $H$, hence $e$ is in $H$
• $x\in H\implies x^{-1}\in H$: Now that $e$ is in $H$, set $b=x,a=e$ to get $x^{-1}e$ is also in $H$, so $x^{-1}$ is in $H$
• $x,y\in H\implies xy\in H$: Set $a=y,b=x^{-1}$. The previous step tells us both are in $H$. So $(x^{-1})^{-1}y$ is in $H$, which tells us that $xy$ is in $H$.

Thus, $H$ satisfies all the three conditions to be a subgroup.