Tamil Nadu Open University (TNOU) 2009-2nd Year B.Sc Mathematics " MODERN ALGEBRA " UG 467 BMS 04 - Question Paper
WS 9
UG-467 BMS-04B.Sc. DEGREE EXAMINATION JANUARY 2009.
(AY - 2005-06 and CY - 2006 batches only)
Second Year Mathematics MODERN ALGEBRA Time : 3 hours Maximum marks : 75
SECTION A (5 x 5 = 25 marks)
Answer any FIVE questions.
1. If the Function f.RR is given by fx2 and g : RR is given by j= sinx, find
and and show that they are not equal.
2. If H and K are subgroups of a given group G, then prove that H r\K is also a subgroup of G.
4. If n is any integer and n j= 1 then prove that
1 <iiod n.
5. Prove that every subgroup of an abelian group is a normal subgroup.
6. If /: GG' is a homomorphism then show that f is one to one if and only if ker / =
7. Prove that the set of all matrices of the form
f a i)\
where a,b&R is a ring under matrix addition
-b a) and multiplication.
8. If R is a commutative ring with identity then prove that R is an integral domain if and only if cancellation law is valid in R.
SECTION B (5 x 10 = 50 marks)
Answer any FIVE questions.
9. If A and B are two subgroups of a group G then prove that AB is a subgroup of G if and only if AB - BA.
10. If G is a group and a e Gthen prove that the order of a is the same as the order of the cyclic group generated by a.
11. Prove that a group G has no proper subgroups if and only if it is a cyclic of prime order.
12. If a group G has exactly one subgroup H of given order then prove that H is a normal subgroup of G.
13. Prove that any finite group is isomorphic to a group of permutations.
14. State and prove the fundamental theorem of homomorphism.
15. Prove that any finite integral domain is a field.
16. Prove that any integral domain D can be embedded in a field F and every element of F can be expressed as a quotient of two elements of D.
3 UG-467
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