Download OU B.Sc 2018 May-June 2nd Year 2nd Semester 7114 Mathematics Question Paper

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Code No: 7114/E/R

FACULTY OF SCIENCE

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B.Sc. IV - Semester (CBCS) Examination, June 2018

Subject: Mathematics

Paper: IV Algebra

Time: 3 Hours Max. Marks: 80

SECTION - A (5 x 4 = 20 Marks)

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(Short Answer Type)

Note: Answer any Five of the following questions

1. Write all subgroups of the group Z30 and indicate their orders.
2. For n>1, show that the alternating group A_n has order n!/2
3. If G is a group and H is a subgroup of index 2 in G, then show that H is a normal subgroup of G.

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4. If G is an abelian group and H is a normal subgroup of G then show that G/H is also an abelian group.
5. Define idempotent element in a ring R. Find all idempotent elements in the ring (Z₁₀, +₁₀, x₁₀)
6. If I₁ and I₂ are any two ideals in a ring R, then show that I₁ n I₂ is always an ideal of R.
7. If f(x) = 1+2x+3x² , g(x) = 2+3x+4x²+x³ then find f(x)+g(x), f(x).g(x) in the ring Z₅[X].
8. Let R be a commutative ring of characteristic 2. Then show that the mapping f : R ? R defined by f(a) = a² ? a?R is a homomorphism.

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SECTION - B (4 x 15 = 60 Marks)

(Essay Answer Type)

9. (a)(i) Let G be a group, H, K be two subgroups of G. Then show that HK= {hk|h ? H, k ? K} is a subgroup of G. (ii) Let G be a group and a?G is such that o(a) = n then show that o(a^k) = n/gcd(n, k) (where k is a positive integer)

   OR

   (b)(i) Let a =(a₁,a₂,a₃....a_n) and ß =(b₁,b₂,b₃..... b_n )are any two disjoint permutations then show that aß = ßa (ii)Let a,ß?S₆ and a=(1 2 4 5 3 6), ß=(1 4 3 2 5 6) then evaluate aß, aß^-1
10. (a)Let G be a group and a,b ? G and H is a subgroup of G then show that (i) aH = bH ? a^-1b ? H (ii) aH is a subgroup of G ? a ? H.

    OR

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    (b)(i)Let G be a finite abelian group and P be a prime that divides the order of G then show that G has an element of order P.
11. (i) Show that every finite integral domain is a field. (ii) Define characteristics of a ring R with unity. Show that the characteristics of an integral domain is either zero or a prime.

    OR

    (Contd on 2)

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