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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**

** **i**s 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**