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PD-153-S.E.-CV-19M.A./M.Sc. MATHEMATICS (l't Semester)Examination, Dec -2020ADVANCED ABSTRACT ALGEBRA (I): Three Hours] [Maxlmum Marks : 80lMinimum Pass Marks : 29
Note : Answer from both tho Seotions as direoted. The figures in the right-hand margin indicate marks.
Section-A
1. Answer the following questions:- 1x10=10(a) What is the order of alternating group An .(b) Deline odd and even permutations.(c) Define centre ofa group what is the centre ofabelian group.(d) Defrne ntr derived set of a group.(e) Define nth centre of a group.(f) Give an example which show solvable group is not abelian,(g) When two subnormal series are called isomorphic or equivalent.ftr) Give example of nil ideal.[) Write a maximal ideal of Z.[) Give example of prime ideal.2. Answer the following questions:- 2xF10(a) Show that Kernel of a group is normal subgroup.(b) Show that cetrtre is a normal subgroup of a group.(c) Show that every nilpotent group is solvable but cotryerse need not be true.(d) Show that Jordan-Holder theorem implies fundamental theorem of arithmetic.(e) Prove that every nilpotent ideal is necessarly nil.Section-B 12x5=60Answer all questions.3, Show that every group is isomorphic to permutation group,ORDefine inner automorphism and show that inner automorphism is a normal subgroupof automorphism of G.4. Show that if a subgroup II and a quotient group G/g are solvable than G is solvable.ORLet G be the group and G' be its commutator subgroup then show that(a) G' is normal subgroup of G(b) for any normal subgroup H ofGG/g is abelian ifand onty if G' c H5. State and prove Jordan-Holder theorem.ORProve that ifG is a commutative group having a composition series then G is abelian.6. Prove that an ideal M of a commutative ring R with unity is a maximal ideel if andonty if fi is a field.ORIf R is finite nonzero commutative ring then show that every maximal ideal is a prime idealbut conyerse need not be true. .7. State Zorn's lemma and show that ifT is a nonzero ring with unity 1, and I is anideal ofR such that I E ft, then there exist a maximal ideal M ofR such thst c M ,ORGive an example to show that nil ideal does not imply nilpotent.
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