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2 D 10666

Check whether Z x Z is cyclic.
Find order of the permutation (1, 3) (2, 6) (1, 4, 5).

Let ®:G, >G,be a group homomorphism. Show that © (6) =e’ where e and e’ are identity
elements of G, and G, respectively.
Define a Ring.
Give example of an integral domain.
(10 x 3 = 30 marks)
Section B

Answer at least five questions.

Each question carries 6 marks.

All questions can be attended.
Overall Ceiling 30.

Solve x> — 3x4 + 4x9 —4x +4 having the root 1 +i.
Solve the cubic equation 233 — x2 —18x +9=0 whose roots area, b, c with a+b=0.

Find an upper limit of the positive roots of the equation 2x° — 7x* — 5x° + 6x? + 3x - 10 = 0.
Prove that set of all even permutations of S, is a subgroup of S,.

Define * on z by a*b=a-—b. Check whether (Z,*) is a group.

Check whether Z,, is cyclic.

Draw the subgroup diagram 91

Let G, and G, be groups and let ©: G, > G, be a function such that (८ ९) = क (५) फ (8) ण बा
a,b eG. Prove that pis 1—1if and only if ®(x)=e implies that x =e for all xe Gy.

(5 x 6 = 30 marks)

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