IGNOU BMTC-134 June 2025 Previous Year Question Paper

Q1.(a) Define a cyclic group. Give an example of a finite, non-cyclic group. (You don’t need to prove that your example is a group. You have to only prove that it is non-cyclic.) 3

(b) State a criterion for checking whether a non-empty subset H of a group G is a subgroup of G. Use the criterion to check whether : c d 0 is subgroup of the group of 2 x 3 matrices over C under addition. 2

(c) Write the permutation (1384 2)0(269)eS,, in the two-line format. 2

(d) State Lagrange’s theorem. What are the possible orders of subgroups of a group of order 18 ? 2

(e) We define o:R[x] >R by i=0 sujective ring homomorphism or not. If it is, find the kernel of the homomorphism and a minimum set of generators of this ideal. If $ is not a homomorphism, define a homomorphism from R[x] onto R.

Q2.(a) Let a-[? a €GL,

(C). Apply the principle of mathematical induction to on fr 0 show that A*” = , and deduce 0 r" that 8201 _ . for all neN. r" 0 Further, show that, if 0

(A) is finite, then o

(A) is even. 6

(b) Show that «+2 and x+4 are coprime in Q[x]. 2

(c) State the Mod p irreducibility test for a polynomial in Z[x]. Use the test to show that the polynomial x? +18x +27 is irreducible in Q[x].

Q3.(a) Define a reflexive relation, a symmetric relation and a transitive relation on a non-empty set S. Consider the set N and the relation ~, defined by a~b for a,beNif a=b" for some keN. Check whether ~ is reflexive, symmetric and transitive. Is it an equivalence relation ? Justify your answer. 5

(b)

(i) Let G be a group and H be its subgroup. Define a left coset of H in G corresponding to an element aeG. 15

(ii) Let G=S, and H = {1, (1 2 3 4), (1 3) (2 4), (1 4 8 2)}. Compute the left coset aH, where a =

(12). 15

(c) Find all the proper ideals of Z/70Z. Further, which of these are maximal ideals and why ? 5

(d) Give an example, with justification, of a subgroup of Ss which is not normal in$,.

Q4.(a)

(i) Show, by a rough sketch, a rotational symmetry of the character ‘N’ of the English alphabet, specifying the point about which it is rotated and the angle of rotation. 15

(i) Show, by a rough sketch, a reflection symmetry of the character ‘T’ of the English alphabet, specifying the line about which it is reflected. 15

(b) How many elements does the group A, have, for neN,n>2 ? List all the elements of Ay. 4

(c) Check whether or not : is a commutative ring with unity under matrix addition and matrix multiplication. If R is a ring, find char R. If R is not a ring, find the characteristic of any ring containing R.

Q5.(a) Prove that the 3 cycles in A, generate A,, for n>3. 6

(b) Find the quotient field of Z[V5]. 3

(c) Let S= {1, 2, 3, 4}, and * be the binary operation on S defined by a*b=b. Compute the Cayley table for (S, *). Is * commutative ? Is * associative ? Justify your answers.

Q6.(a) State the Unique Factorisation Theorem for polynomials over a field. Further, show that 6x7 +3x+5 factors as (6x +2)(x+6) and as (8x +1) (2x +5) in Z,. Explain why this doesn’t contradict the said theorem. 4

(b) For a, be Z, show that:

(i) aZAbZ=(Z, where ¢=[a, b], the l.c.m. of a and b. Gi) aZ+bZ=hZ, where h = (m, n) is the g.c.d. of a and b. Further, find 3Z.04Z and 6Z+15Z. 7

(c) Show that G/H is an infinite group, where G=GL,(R) and H=SL,(R).

Q7.(a)

(i) Give an example, with justification, of a ring R and an element acR such that a is neither a unit nor a zero divisor of R. 2

(ii) Give an example, with justification, of a ring R and an element acR such that a is a zero divisor of R which is not nilpotent. 2

(b) If is an ideal of a ring R, show that : I[x]= [Ea x' la; elne ४०० i=0 is an ideal of R[x]. Further, will I[x] be a principal ideal if R is a PID ? Justify your answer. 4

(c) Show that : G= " meZ is a subgroup of GL,(R). Further, check whether or not yw:Z—G, defined by (n) Ton is a rou! yi 01 group isomomorphism, where Z is the group of integers under addition. 5

(d) Show that for neN, and for every d|n,D,, has a subgroup of order d.

Q8.Which of the following statements are true, and which are false ? Justify your answers with a short proof or a counter example, whichever is appropriate. Marks will only be given for proper justification : 10 @) Every non-abelian group has at least one proper subgroup which is not normal. Gi) If a group has elements of order three and two, it has an element of order six.

(ii) Every subring of a non-commutative ring is non-commutative.

(iv) If S is a ring and R is a subring of S and S has zero divisors, then R also has zero divisors.

(v) If every element of a group has infinite order, the group has no_ proper subgroups of finite index. fe eq: मर ° sll bc, aec| c d 0 (डछ) o:R[x] oR को o{ San! aya i=0 0 r" r 0 waa N% a~b, abeN ae a=b' 7 में 6 +%+6 (G@x4+2)(x+6) aR

(6) aZnbZ=(Z %, Fe 0 =[a, b] है, जो 6

(ii) aZ+bZ=hZ %, Tet h=(m,n) % Ft fH: i=0 G न mm <7}