Union & Intersection of Subgroups | HK Subgroup Theorem | BSc Maths CSIR NET IIT JAM Hindi

Group Theory mein subgroups ke beech ka relationship samajhna — union, intersection, aur product — directly IIT JAM aur CSIR NET mein 2–3 marks dilata hai. Aaj ka lecture ek very important theorem cover karta hai: HK subgroup hoga agar aur sirf agar HK = KH. Yeh condition bahut students miss kar dete hain aur exam mein galat answer dete hain!
📚 Is Video Mein Cover Hoga:

Union of two subgroups — kab subgroup hota hai, kab nahi
Intersection of subgroups — theorem aur proof
Complex product HK — definition aur properties
HK is a subgroup iff HK = KH — complete proof
Congruence relation modulo a subgroup — definition aur examples
Exam-style solved problems — IIT JAM pattern

🎓 Yeh Video Kin Ke Liye Useful Hai:

IIT JAM Mathematics
CSIR NET Mathematical Sciences
GATE Mathematics
BSc / MSc Maths students

⏱️ TIMESTAMPS:
0:00 - Introduction and recap of Subgroups and the One-Step Test
1:43 - Introduction to Operations on Subgroups: Union and Intersection
3:43 - Theorem 1: The Intersection of two subgroups is always a subgroup
5:09 - Step-by-step proof for the Intersection of subgroups
8:20 - Note: Intersection of multiple/infinite subgroups is also a subgroup
9:15 - Summary of Intersection with an example (2Z ∩ 3Z = 6Z)
10:20 - Introduction to the Union of Subgroups
10:36 - Theorem 2: Union of two subgroups is a subgroup if and only if one is contained in another
12:03 - Necessary Condition proof for the Union of subgroups
14:39 - Sufficient Condition proof for the Union of subgroups
23:02 - Theorem 3/Counter-example: Union of subgroups is generally not a subgroup
23:47 - Step-by-step counter-example proof using 4Z and 5Z subgroups
27:16 - Visual summary and recap of Union vs. Intersection of subgroups
29:05 - Theorem 4: The complex HK is a subgroup iff HK = KH
30:31 - Necessary condition proof for HK = KH
36:23 - Sufficient condition proof for HK as a subgroup
41:26 - Theorem 5: Left/Right Congruence Modulo H is an Equivalence Relation
43:48 - Proof of Reflexive property for Congruence Modulo H
44:54 - Proof of Symmetric property for Congruence Modulo H
46:20 - Proof of Transitive property for Congruence Modulo H
48:46 - Summary of the lecture and closing remarks

🔗 Previous Lecture:
Subgroups — Definition, Complex, One-Step & Two-Step Test → https://youtu.be/ODxs0zHZHJ8
📺 Complete Playlist:
https://www.youtube.com/playlist?list=PLHoWYh3APMj5bs8M2Uf0sonXo3zi5wDZV

📢 CONNECT WITH ME:
📲 Telegram ▶ https://t.me/fractalfrontiermaths
📸 Instagram ▶ https://www.instagram.com/mathsworld007
🌐 VISIT MY WEBSITE: https://drpraveendra.org/
📝 LATEST BLOG POST: https://drpraveendra.org/blog

📚 DR. PRAVEENDRA SINGH’S RECOMMENDED REAL ANALYSIS BOOKS
Curated from years of teaching and research experience at IIT Roorkee. 🎓

For a complete list of my top-rated mathematics books across all subjects, visit my recommendation page here: 👉 https://drpraveendra.org/recommended-books

⭐ 1. Contemporary Abstract Algebra (11th Edition) By Joseph A. Gallian
👉 https://amzn.to/4cggqzS

⭐ 2. Abstract Algebra (3rd Edition) By David S. Dummit & Richard M. Foote
👉 https://amzn.to/4msSEDW

⭐ 3. A Course in Abstract Algebra (5th Edition) By V. K. Khanna & S. K. Bhambri
👉 https://amzn.to/4tPWqJZ

⭐ 4. A First Course in Abstract Algebra (7th Edition) By John B. Fraleigh
👉 https://amzn.to/3Qqs0j8

⭐ 5. Abstract Algebra Book — CSIR NET Reference
👉 https://amzn.to/4tqkJ1i

#SubgroupTheorems #IITJAMMaths #BScMaths #HindiMaths #FractalFrontierMaths

Видео Union & Intersection of Subgroups | HK Subgroup Theorem | BSc Maths CSIR NET IIT JAM Hindi канала Fractal Frontier Maths