The Intersection of Two Subgroups is also a Group Proof  Summary and Q&A
TL;DR
The intersection of subgroups in a group is itself a subgroup.
Key Insights
 🤬 The symbol for "subgroup" is denoted by ⊆.
 🇭🇰 The intersection of subgroups H and K is denoted as H ∩ K.
 🇭🇰 Closure under the group operation means that the product of two elements in H ∩ K remains within H ∩ K.
 🇭🇰 Closure under inverses ensures that the inverse of any element in H ∩ K also belongs to H ∩ K.
 🇭🇰 By satisfying the conditions of nonempty intersection, closure under the group operation, and closure under inverses, H ∩ K is confirmed to be a subgroup of G.
 ❓ Understanding the proof requires grasping the concept of subgroups, the closure property, and the definition of intersection.
 👥 The proof establishes a fundamental result in group theory regarding subgroups and their intersections.
Transcript
Read and summarize the transcript of this video on Glasp Reader (beta).
Questions & Answers
Q: What does the symbol for "subgroup" represent?
The symbol for "subgroup" indicates that a certain subset of a group follows the same group structure and contains the identity element and inverses.
Q: How is it shown that the intersection of H and K is not empty?
Since H is a subgroup of G, it contains the identity element (e), and likewise, K being a subgroup of G also contains e. Therefore, the intersection must contain e as well, making it nonempty.
Q: How is closure under the group operation proven for H ∩ K?
If X and Y are elements in both H and K, then by the closure property of H and K, the product XY is also an element of H and K. Hence, XY belongs to the intersection, demonstrating closure under the group operation.
Q: How is closure under inverses shown for H ∩ K?
Assume X is an element in H ∩ K. By definition, X is in H and K. As H and K are closed under inverses, the inverse of X exists in both H and K. Thus, the inverse of X resides in H ∩ K, proving closure under inverses.
Summary & Key Takeaways

The content explains a proof showing that if H and K are subgroups of a group G, then their intersection (H ∩ K) is also a subgroup of G.

The proof consists of three steps: showing that the intersection is not empty, proving closure under the group operation, and demonstrating closure under inverses.

By satisfying these three conditions, it is proven that H ∩ K is indeed a subgroup of G.