BSc Mathematics SEM VI 2016 17 2016-17 Algebra II Question Paper - Mumbai University | munotes

Questions asked in this paper

• Figures to the right indicate marks for respective

1. Q1 (a) Answer any ONE
   • i. State and prove the first isomorplisin for 8
   • ii. H be of G, then prove that (8) 4 if HaHb = Hab for all G Answer any TWO : i, If Hy, Hyare normal subgroups of Gy, then x Hy (6) a is normal in x G2 Show that any two cyclic groups-of same order (6) 4
   • iii. Let f : G’ be a group honiomorphism. Prove that if in G’ then is normal 6
   • iv. Show that {e, b} is but not normal in ab, 6
2. Q2 (a) Answer any ONE :
   • i. Show that field. Give example-of infinite integral domain example of a finite ring that is not an 8
   • ii. Define of ring. ‘Let 1g be the multiplicative identity of R. Show that characteristic m only if order-of Lz in the group (R, +) is otherwise show that the characteristic of integral domain is 8
   • (b) Answet any
   • i. R; commutative rings and f be an onto homomorphism. is.an ideal of show that f(Z) ideal of A’ ii, Show that the ina are zero ideal and F itself. (6) Pind the of + C defined by =~ (6) iv, Show that a field has characteristic 2. (6) ‘Show that any prime: element of an integral domain is also an irreducible (8) Further-show that the converse is true in PID 6
   • ii. Define an’ Euchdean domain. Show that every Euclidean domain is a PID. 8

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