BSc Mathematics SEM IV 2018 19 Apr 2018-19 MATHEMATICS PAPER II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 Choose correct alternative in each of the following 20 marks
   • i. Let where a and b denotes rotation and reflection then =
   • (c) 6 (d) None of the above
   • ii. Let Hand K be the subgroups of a group G. Then HUK
   • (a) Is always a subgroup of G
   • (b) Is subgroup of G
   • (d) None of the above
   • iii. Z, group under the binary operation
   • (c) (d) None of the above
   • iv. Inthe group order of 10 is
   • v. Let H isa proper subgroup of Z under addition and 12, 14, 18 H then
   • vi. Let a be an element of a group G and let order of a in G be infinite then how many generators does the group < a > have?
   • vii. If G = (Z,+) = {0,+3, +6, then 7+H=23+H (d) None of these
   • viii. Let G bea group of order 8 then G must have an element of order
   • (c) 8 (d) None of these
   • ix. Let — C* given by (x) = x* be ahomomorphism then =
   • (c) (d) None of these
   • x. Let G be an abelian group which has no element of order 2 and @:G —G given by = x?, then
   • (a) an automorphism group homomorphism which may not be one —one an automorphism if G is finite is not a group homomorphism
2. Q2 Attempt any ONE question from the following: Show that form a group under the Define Centre of Group G. Hence or otherwise prove that the Centre of any group is a subgroup of the group 8 marks
3. Q2 Attempt any TWO questions from the following: 12 marks
   • b) i. Let G be a group. Prove that Vabe
   • ii. Let G be a group and a G. Show that H = Z} is a subgroup
   • iii. Let Gbe a group and a G with O(a) = n then show that if and only if
   • iv. B=(134)(265)( 23 4) .Write and B as a product of disjoint cycles. Further, verify the following
   • q) = o(Ba)
4. Q3 Attempt any ONE question from the following: 8 marks
   • a) i. Prove that Z, the set of residue classes modulo n is a group under addition. Also determine all the generators for il. Let G be a finite cyclic group of order n then prove that G has a unique subgroup of order d for every divisor d of n
5. Q3 Attempt any TWO questions from the following: 12 marks
   • b) 1. Let be a finite cyclic group of order 12 then what are all the generators of G. Also determine all the generators of the subgroup H =
   • ii. Determine all the subgroups of the cyclic group Zj, Show that H = z} is a cyclic subgroup of
   • iv. Consider the set {4,8,12, 16}. Show that this set is a group under multiplication modulo 20 by constructing a Cayley table. What is the
6. Q4 Attempt any ONE question from the following: 8 marks
   • a) i. Let H is a subgroup of a group G then aH = H if and only if H Further aH is subgroup of G if and only if a H
   • ii. Let is onto group homomorphism. Prove that
   • (p) If H is subgroup of G then f(H) = {f(h)/h H} is subgroup of G’
   • (q) If is subgroup of G’ then = {a G/f(a) H’} is subgroup of G and kerf
7. Q4 Attempt any TWO questions from the following: 12 marks
   • b) i. State Lagrange's theorem for finite group. If H and K are subgroups of G such that o(H) = 12 and o(K) = 35 then show that HN K = {e}
   • ii. Let G bea finite group then show that Show that f:G — G given by f(x) = x7? is a automorphism if and only if G is abelian Show that G = {a + Q} and H = /a,be Q} are isomorphic groups under addition
8. Q5 Attempt any FOUR questions from the following: Construct composition table of Zé under multiplication modulo 5. Also find the order of each of its elements 20 marks
   • b) Define Abelian group. If (ab)? = a*b? for every a,b in a group G, show that
   • c) Prove that a group of order 3 must be cyclic
   • d) group and let ‘a’ be an element of G =e , what can you say about order of a
   • (ii) Suppose that G is cyclic and o(G) = 24 . Further if # e and # e then show that < a >=G
   • e) Give an example of a group G and a subgroup H of G such that aH = bH but
   • f) Find the number of group homomorphism from to

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