Normal Subgroup | Cosets, Theorems & Klein Four Group Examples | B.Sc. 1st Year Maths

Описание: Normal Subgroup | Cosets, Theorems & Klein Four Group Examples | B.Sc. 1st Year Maths


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In this video, we study one of the most important topics of Group Theory — Normal Subgroups, designed strictly according to the B.Sc. 1st Year syllabus of Introduction to Algebra and Number Theory.

This lecture covers cosets, definition of normal subgroup, important theorems, normal subgroup tests, and solved problems, including detailed examples from the Klein Four Group (K₄).

The explanation is step-by-step, exam-oriented, and conceptually clear, making it ideal for college students and competitive exam aspirants.


⏱️ Time Stamps (Chapter-wise Explanation)
🕒 0:00 – Introduction
🕒 0:28 – Cosets: Definition of left and right cosets with explanation and examples.
🕒 3:24 – Normal Subgroup: Definition of normal subgroup, notation, and conceptual meaning.
🕒 5:24 – Theorem -1 & Theorem -2(Equivalent conditions for normal subgroups and Normal Subgroup Test.)
🕒 8:58 – Theorem -3 Every subgroup of an abelian group is normal (with proof).
🕒 10:32 – Theorem -5(A subgroup is normal if the product of two right cosets is again a right coset.)
🕒 12:58 – Problem 1 (Klein Four Group)
Normal subgroups of the Klein Four Group
🕒 15:31 – Problem 2 (Another solved problem related to normal subgroups.)
🕒 17:38 – Problem 3 (Proof that the center of a group is a normal subgroup.)
🕒 18:48 – Outro


🎯 What You Will Learn in This Video
✔ Definition of cosets
✔ Left coset vs right coset
✔ Definition of normal subgroup
✔ Normal subgroup tests
✔ Important theorems with proofs
✔ Solved problems on Klein Four Group
✔ Center of a group as a normal subgroup
✔ Exam-oriented explanations


🎓 Who Should Watch This Video?
B.Sc. 1st Year Mathematics students
Students studying Introduction to Algebra & Number Theory
CSIR-NET / GATE / JAM aspirants
Anyone learning Group Theory from basics


📌 Syllabus Covered
B.Sc. Mathematics – Paper: Introduction to Algebra and Number Theory
Unit: Group Theory (Cosets & Normal Subgroups)


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