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SECTION- A Answer ALL the questions (10x2=20) 1. Define a partially ordered set and give an example. 2. If G is group then prove that the identity element of G is unique. 3. Show that every subgroup of an abelian group is normal. 4. Define order of an element in a group. 5. Express (1,3,5) (5,4,3,2)(5,6,7,8) as a product of disjoint cycles. 6. Define an automorphism of a group with an example. 7. Define an integral domain and give an example. 8. If F is a field, prove that its only ideals are {0} and F itself. 9. Define maximal ideals. 10. Find all the units in 𝑍[𝑖] = {𝑥 + 𝑖𝑦 ∶ 𝑥, 𝑦 ∈ 𝑍}. SECTION – B Answer any FIVE questions (5x8=40) 11. If H and K are subgroups of G, then prove that HK is a subgroup of G if and only if HK=KH. 12. Prove that every subgroup of a cyclic group is cyclic. 13. Prove that a subgroup N of a group G is a normal subgroup of G if and only if the product of two left cosets of N in G is again a left coset of N in G. 14. If f is a homomorphism of a group G into a group G then prove that kernel of f is a normal subgroup of G. 15. Prove that every group is isomorphic to a group of permutations. 16. Let Q be the set of all rational numbers and let 𝑄(√2 ) = {𝑎 + 𝑏√2: 𝑎, 𝑏 ∈ 𝑄}. Show that 𝑄(√2 ) is a field under usual addition and multiplication. 17. Prove that every Euclidean ring is a principal ideal domain. 18. Let R be a commutative ring with unity and P an ideal of R .Then prove that P is a prime ideal of R if and only if R/P is an integral domain. SECTION – C Answer any two questions 19. a.) If H and K are finite subgroups of a group G then prove that 0(𝐻𝐾) = (2x20=40) 0(𝐻)0(𝐾) 0(𝐻∩𝑘) 𝑛 b.)If 𝑎 ∈ 𝐺 and 𝑎 = 𝑒, prove that 0(a) divides n. 20. a.) State and prove Lagrange’s theorem. b.) If H and K are two subgroups of a finite group G and 𝐻 ⊆ 𝐾 , show that [𝐺: 𝐻] = [𝐺: 𝐾][𝐾: 𝐻]. (12 + 8) (12 + 8) 21. a.) State and prove Fundamental theorem of homomorphism of a group G. b.) Show that an arbitrary intersection of ideals of a ring R is an ideal of R. (12 + 8) 22. a.)Let R be a Euclidean ring. Then prove that every non-zero element of R is either a unit in R or can be uniquely written as a product of a finite number of prime elements of R. b.) Let R be a Euclidean ring. Then prove that any two elements a and b in R have a greatest common divisor d which can be expressed in the form 𝜆𝑎 + 𝜇𝑏 for some 𝜆, 𝜇 ∈ 𝑅. (12 + 8) $$$$$$$