Msc First Semester Maths (Advanced Algebra) Question Paper 2025 Pdf Download : Msc 1st Semester Maths Question Paper 2025 Pdf
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Paper Code: 41151
B-1501
M.Sc. Mathematics (Semester First) Examination, 2025
Advanced Algebra
Time: Three Hours ] [Maximum Marks: 70
Note: Attempt all the sections as per instructions.
Section-A
(Short Answer Questios)
Note: Attempt any four que ons out of the following. Each questions carries 07 marks.
1. Prove that a non-empty subset H of a group G is a subgroup of G if and only if ab’eH va, beH.
2. Prove that every group is isomorphic to a group of permutations.
3. The set of automorphisms of a group and the set of inner automorphisms of a group are both groups under the operation of function composition. Prove it.
4. Every group of order p² (where p is a prime number) is abelian. Prove it.
5. Let R be a commutative ring with unity and let A be an ideal of R. Then prove that R/A is a field if and only if A is maximal.
6. State and prove Schwartz’s inequality.
7. Show that x-3 is solvable by radicals over the field of rationaes(Q).
8. Let N be a normal subgroup of a group G. If both N and G/N are solvable, then prove that G in also solvable.
MSc 1st Semester Maths (Advanced Algebra) Question Paper 2024-25 Pdf
Section-B
(Long Answer Questions)
Note: Attempt any three questions out of following. Each question carries 14 marks.
9. Prove that every principal ideal Domain is a unique factorization domain.
10. Let H be a subgroup of a finite group G and H is a power of a prime p, then prove that H is contained in some sylow p-subgroup of G.
11. State and prove Cauchy theorem in group theory.
12. Let & be a group homomorphism from G to G Then prove that the mapping from G/Ker & to $(G) is an isomorphism.
13. Let IF be a field of characteristic zero and let aef If E is the splitting field of x-a over IF, then prove that the Galois group Gal (E/F) is solvable.
14. Let R be a ring and let f:MM’ be a left R-module homomorphism. If N is a sub module of M contained in Ker(f), then prove that there exists an R-module homomorphism F: M/N M’ such that→ (i) im (f) im (f) (ii) ker(f) kerf/N
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