Group Theory Question Paper 2025

Four Years B.Sc. (Honours) (CBCS) Degree Examination – October/November 2025

Students appearing for the Third Semester Mathematics examination can download the Group Theory Question Paper 2025 from this page. This question paper is useful for students preparing for Mathematics examinations and for practising important Group Theory problems.

Group Theory 2025 Question Paper – Section A

1. If H is any subgroup of a group (G, *) and h ∈ G, then prove that h ∈ H if and only if hH = H = Hh.
2. Show that every subgroup of an abelian group is normal.
3. Prove that the intersection of any two normal subgroups of a group G is a normal subgroup.
4. Prove that any two left (right) cosets of a subgroup are either identical or disjoint.
5. If f is a homomorphism of a group G into a group G′, then prove that the kernel of f is a normal subgroup of G.
6. Find the regular permutation group isomorphic to the multiplicative group {1, −1, i, −i}.

Note: These Section-A questions cover important concepts including subgroups, normal subgroups, cosets, group homomorphisms, and permutation groups.

Group Theory 2025 Question Paper – Section B

UNIT – I

1. Prove that a finite semigroup G satisfying cancellation laws is a group.

2. If * is defined on ℤ as a * b = a + b + 1, for all a, b ∈ ℤ, then show that (ℤ, *) is an abelian group.

UNIT – II

3. State and prove Lagrange’s theorem.
4. Prove that the union of two subgroups of a group G is a subgroup if and only if one is contained in the other.

UNIT – III

5. Prove that a subgroup H of a group G is a normal subgroup of G if and only if the product of two right (left) cosets of H in G is again a right (left) coset of H in G.
6. If M, N are two normal subgroups of G such that M ∩ N = {e}, then show that every element of M commutes with every element of N.
7. State and prove the Fundamental Theorem of Homomorphism of groups.
8. If f is a homomorphism from a group G into a group G₁, then prove that Ker f is a normal subgroup of G.

UNIT – IV

9. If f = (12345876), g = (41587328), prove that (fg) = gf.
10. State and prove Cayley’s theorem.

UNIT – V

The Unit-V question at the end of the supplied PDF is partially cut off in the available text extraction. Please refer to the original PDF for the complete question.

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Use the original PDF to read the complete question paper format, including any missing portions.

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Why Practise Group Theory Previous Papers?

• Helps understand the exact VSU examination pattern and structure.
• Identifies frequently tested concepts such as subgroups, cosets, and homomorphisms.
• Provides targeted practice with both short-answer and long-answer formatting.
• Serves as an excellent revision tool before the final Mathematics examination.

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