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IGNOU MTE-6: ABSTRACT ALGEBRA June, 2017 QUESTION PAPER
Note: Attempt Five questions in all. Question no. 7 is compulsory. Answer any four questions from questions no. 1 to 6. Use of calculators is not allowed.
Q1. (a) Let X be the set of all lines in R × R. Consider the relation ‘~’ on X given by ~ if and only if is perpendicular to . Is the relation ‘~’ reflexive, symmetric or transitive? Justify your answers.
(b) If I is a non-trivial ideal in a field F, then check whether I = F or not.
(c) Let . Prove
that R is a ring with respect to matrix addition and matrix multiplication.
Ans. FOR FULL ANSWER
Q2. (a) Express as a product of disjoint cycles, where
Is it an even permutation? Justify your answer.
Ans. Same as June-2007, Q.No.-3(b) FOR FULL ANSWER
(b) Find the nil radical of .
Ans. Same as Chapter-4, Q.No.-13 FOR FULL ANSWER
(c) Check whether the polynomial is irreducible over or not. Is it irreducible in Z[x] also? Why? Check whether or not is a field.
Q3. (a) Show that is not a unique factorisation domain by expressing 4 as a product of two irreducible elements in R in two different ways.
(b) Find the remainder obtained on dividing by 11.
(c) Find the maximal domain possible and corresponding range of the function f, defined by f(x) = .
Ans. FOR FULL ANSWER
Q4. (a) Show that every group of order 44 has a proper non-trivial normal subgroup.
(b) Take
which is a group under matrix multiplication. Check whether
is a subgroup of G. If it is a subgroup, then check whether it is a normal subgroup in G. If H is not a subgroup of G, obtain a proper non-trivial subgroup of G.
Ans. Refer to Chapter-3, Q.No.-12 FOR FULL ANSWER
(c) Let R be a commutative ring and let aR. Show that
is an ideal of R.
Q5. (a) Find all the generators of a cyclic group of order 12.
(b) Show that the map defined by mod 2, is an onto ring homomorphism. Obtain Ker f, and check whether or not this is a maximal ideal of
Ans. Same as Chapter-3, Q.No.-10 & Refer to Page No.-142, Chapter-14(d) [Important Question] FOR FULL ANSWER
(c) Write down all the elements of the quotient group Is any element of order 5? Give reasons for your answer.
Q6. Which of the following statements are true? Justify your answers.
(a) {–p, IGNOU, Australia, Z) is a set.
(b) The characteristic of a field containing (50 –1) elements is 50.
(c) If H is a subgroup of a group G such that = p, where p is a prime number, then
(d) Every subring of a non-commutative ring is non-commutative.
(e) If (G, •) is a group, then f : G × G ® G, defined by f (g, h) = is a binary operation on G.
Ans. FOR FULL ANSWER
Q7. (a) Let D be a Euclidean domain and d be the Euclidean valuation on D. Show that if a and b are associates in D, then d(a) = d(b).
Ans. Refer to Page No.-142, Q.No.-13 [Important Question] FOR FULL ANSWER
(b) Take X = a subset of Form the Cayley table of X with respect to multiplication modulo 40. Check whether X is a group or not, with respect to the given operation.
Ans. Same as Chapter-4, Q.No.-14 FOR FULL ANSWER
(c) Let G be the group of quaternions. Find Z(G), and hence,
Ans. Refer to Gullybaba.com “download section” FOR FULL ANSWER
