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

Questions asked in this paper

• (2) Figures to the right indicate marks for respective subquestions

1. Q1 (a) Answer any ONE Let H be a subgroup of group G. Prove thatthe following statements are: (8)
   • (q) for each a
   • (r) Every left coset of H in G is also a right H in G ie. aH = Ha
   • ii. State and prove the Cayley’s theorem for finite group. 8
   • (b) Answer any TWO
   • i. normal subgroups of G and H be a subgroup of Prove that 6
   • ii. If a cyclic group H of a group G is normal in G, then show that every subgroup of H is normal in G 6
   • iii. Find the order of each element of Zy x Is x isomorphic to Zs? 6
   • iv. Suppose G is a non-abelian group of order p? where a prime and Z {e}, then prove that |Z(G)| = p 6
2. Q2 (a) Answer any ONE
   • i. Let be commutative rings and bea ring homomorphism. 8
   • (p) If f is surjective, J is an ideal of R, then is an ideal of R’
   • (q) If J’ is an ideal of R’ , then is an ideal of R
   • ii. Show that, characteristic of a ring R is n if and only if the order of the multiplicative identity of R is n in the group (R, +). Further if char R = n, where is an integral domain, then show that n is a prime 8
   • (b) Answer any. TWO
   • i. Let A be a subring and B be an ideal of a ring R. Then prove that AN B is an ideal of A and A/(AN B) (A+ B)/B 6
   • i. Let R be a commutative ring. Show that J = for some n N} is an ideal of R. Also show that R/J has no nilpotent element iti. Show that there is exactly one non-zero ring homomorphism from (6) 6
   • iv. Show that, if R is a ring having 6 elements then R is commutative. Is R an 6

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