12-18-2014, 09:25 PM

Part A( Short Answer Type Questions)

Answer all the questions

(Each Questions has weightage one)

Time: 3Hours Max. Weightage: 36

1. Define Hamming weight wt(u) of a word u and distance d(u, v) between two binary words u and v in Bn.

Let u = 1101010111 and v = 0111001110. Find (i)wt(u), (ii)wt(v), (iii)d(u, v).

2. Find the order of the element (3, 6, 12, 16) in Z4 × Z12 × Z20 × Z24.

3. Find all abelian groups, upto isomorphism of order 1089.

4. Let ϕ : Z18 → Z12 be the homomorphism such that ϕ(1) = 10. Find the kernel K of ϕ. Write the cosets of

Z18/K and list the elements in each coset.

5. Show that if a finite group G contains a proper subgroup of index 2 in G, then G is not simple.

6. Find an isomorphic refinement of the series

{0} < 8Z < 4Z < Z and

{0} < 9Z < Z.

7. Show that Z has no composition series.

8. Show that S3 is solvable.

9. Show that if H and K are normal subgroups of a group G such that H ∩ K = {e}, then hk = kh for all

h ∈ H and k ∈ K.

10. Show that no group of order 20 is simple.

11. Compute (i + 3j)(4 + 2j − k) in the Quaternions Q.

12. If F is a field and a ̸= 0 is a zero of f(x) = a0 + a1x + a2x

2 + · · · + anx

n in F[x], show that 1

a

is a zero of

an + an−1x + an−2x

2 + · · · + a0x

n.

13. Factorize x

4 + 4 in Z5[x].

14. Show that the ring M2® of all 2×2 matrices with entries in R, the field of real numbers, is not isomorphic

to the field C of complex numbers.

Part B ( Paragraph Type Questions)

Answer any Seven questions

(Each question has weightage two)

15. Show that Bn with the operation of word addition is a group.

16. Show that the group Zm × Zn is isomorphic to Zmn if and only if m and n are relatively prime.

17. Prove that M is a maximal normal subgroup of a group G if and only if G/M is simple.

18. Compute the factor group Z4 × Z6/ < (2, 3) > .

19. Find the number of distinguishable ways the edges of an equilateral triangle can be painted if four different

colours of paint are available, assuming only one colour is used on each edge and the same colour may be

used on different edges.

20. Obtain the class eqution of D4.

21. Show that for a prime number p, every group G of order p2is abelian.

22. State and prove second isomorphism theorem.

23. Define group presentation. Write a presentation of Z6.

24. Show that the multiplicative group of all non-zero elements of a finite field is cyclic.

Part C (Essay Type Questions )

Answer Any Two Questions

(Each Question has weightage Four )

25. (a) Let X be a G−set and let x ∈ X. Show that |Gx| = (G : Gx).

(b) Find the number of orbits in {1, 2, 3, 4, 5, 6, 7, 8} under the subgroup of S8 generated by < (1, 3, 5, 6) > .

26. (a) State and prove the third Sylow theorem.

(b) Show that a group G of order 30 is not simple.

27. (a) State and prove Eisenstein condition for irreducibility.

(b) Show that x

4 − 2x

2 + 8x + 1 is irreducible over Q.

28. (a) Give the addition and multiplication tables for the group algebra Z2(G), where G = {e, a} is a cyclic

group of order 2.

(b) Let H be a subring of the ring R. Prove that multiplication of additive cosets of H is well defined by

the equation (a + H)(b + H) = ab + H if and only if ah ∈ H and hb ∈ H for all a, b ∈ R and

h ∈ H.