Abstract Algebra 2024 – BSc Computer Science Part 2

Total No. of Questions : 9] [Total No. of Printed Pages: 4

Time: Three Hours]

[Maximum Marks: 50

Note: Attempt five questions in all selection at least one question from each section. All questions carry equal marks.

Section-A

1. (a) Prove that this inverse elements of a group is unique.

(b) Show that the set of invertible 2 x 2 matrices with real entries, with operation given by matrix multiplication is a group.

2. (a) Prove that the center of a group G is a subgroup of G.

(b) Find all generators of Z₆, Z₈, and Z₂₀.

3. (a) Prove that a finite group is the union of proper subgroup iff the group is not cyclic.

(b) What are even and odd permutations? Prove that out of n! permutations, n/2! are even permutations and n/2! are odd permutations.

Section-B

4. (a) State and prove Lagrange’s Theorem.

(b) Let G be a group and let Z(G) be the center of G, then show that, if G/Z(G) is cyclic, G is Abelian.

5. (a) State and prove unique factorization theorem.

(b) Show that the nilpotent elements of a commutative ring a form a subring.

6. (a) Let R be a commutative ring with unity and let A be an ideal of R, then prove that R/A is a field iff A is maximal.

(b) Determine all ring isomorphisms from Z_n to itself.

Section-C

7. (a) Show that if F is a field, then F[x] is a principle ideal domain.

(b) Prove that the intersection of two subspaces of a vector space is also a vector space.

8. (a) If (u, v, w) is a linearly independent subset of a vector space, show that {u, u+v, u+v+w} is also linearly independent.

(b) If f is a homomorphism of U(F) into V(F), then prove that :

(i) f(Ō) = O’, where Ō and O’ are the zero vectors of U and V respectively.

(ii) f(-α) = -f(α) : ∀ α ∈ U

9. (a) Let V = R³ and W={(a, b, c) ∈ V |a² + b² = c²}. Is W a subspace of V? if so, what is its dimensions?

(b) Let $V = \begin{Bmatrix} \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} | a, b, c \epsilon Q \end{Bmatrix}$ Determine a basis for V over Q.

……..End……..

Thank you!