To prove that $S$ is a subgroup of $G$ if and only if $S$ is non-empty, and for all $x, y \in S$, we have $x^{-1}y \in S$, we'll prove this in two parts: necessity and sufficiency.

Part 1: Necessity (Subgroup $\Rightarrow$ Condition)

Assume that $S$ is a subgroup of $G$. We need to show that $S$ is non-empty and for all $x, y \in S$, $x^{-1}y \in S$.

1. Non-empty: Since $S$ is a subgroup of $G$, it must contain the identity element $e$. Therefore, $S$ is non-empty.

2. For all $x, y \in S$, $x^{-1}y \in S$: Since $S$ is a subgroup, it must be closed under the group operation and inverses. This means that for any $x \in S$, the inverse $x^{-1} \in S$, and for any $y \in S$, the product $x^{-1}y$ must also be in $S$. Therefore, $S$ satisfies the given condition that for all $x, y \in S$, $x^{-1}y \in S$.

Hence, if $S$ is a subgroup of $G$, then $S$ is non-empty and for all $x, y \in S$, $x^{-1}y \in S$.

Part 2: Sufficiency (Condition $\Rightarrow$ Subgroup)

Now, assume that $S$ is non-empty and that for all $x, y \in S$, $x^{-1}y \in S$. We need to prove that $S$ is a subgroup of $G$. To do this, we need to verify the subgroup criteria:

1. Identity element: Since $S$ is non-empty, take any element $a \in S$. Consider the case where $x = y = a$. Then, by the given condition, $a^{-1}a = e \in S$. Therefore, $S$ contains the identity element $e$.

2. Closure under inverses: Let $x \in S$. Since $e \in S$, by the given condition, $x^{-1}e = x^{-1} \in S$. Therefore, $S$ is closed under taking inverses.

3. Closure under the group operation: Let $x, y \in S$. We need to show that $xy \in S$. Since $y \in S$ and $x^{-1} \in S$, by the given condition, $(x^{-1})^{-1}y = xy \in S$. Hence, $S$ is closed under the group operation.

Since $S$ contains the identity element, is closed under taking inverses, and is closed under the group operation, $S$ is a subgroup of $G$.

Conclusion:

Thus, $S$ is a subgroup of $G$ if and only if $S$ is non-empty and for all $x, y \in S$, $x^{-1}y \in S$.

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Would you like further details or have any questions?

Here are 5 related questions for deeper exploration:

1. What are some examples of subgroups in well-known groups like $\mathbb{Z}$ or $\mathbb{R}^*$?
2. How can you check if a given subset of a group is a subgroup using the two-step subgroup test?
3. How does the property $x^{-1}y \in S$ relate to the concept of cosets in group theory?
4. Can the condition $x^{-1}y \in S$ be interpreted geometrically for certain groups, such as matrix groups?
5. How does this subgroup criterion simplify in the case of finite groups?

Tip: When dealing with subgroups, remember that the identity element is always a part of any subgroup, and this is often a quick way to check for non-empty subsets.