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2 D 30568
Find the order of the permutation (1, 2, 5) (2, 3, 4) (5, 6).
1.6 © = 219, and let H be the subgroup 4Z 19. Find all cosets of H.

Define normal subgroup of a group G. Give an example.

Ze x Z4

(2.2) '

Compute the factor group

Define commutative ring. Give an example.
Define Integral Domain. Give an example.
Section B

Answer any number of questions.
Each question carries 5 marks. Ceiling is 35.

If (a, n) = 1, then show that a") =1 (1706 7).

Let G be a group and let H be a subset of G. Then show that H is asubgroup of Gif and only if H

isnon-empty and ab! < H foralla,beH.

Let Gbe a finite cyclic group with n elements. Show that G < ‏.ہر‎

Let १: ७1 > Gy bea group homomorphism with Ker ம்‌ - {x € 061 : ९ (४) ८ 2. Show that is one to
one if and only if Ker = {e}.

Let G be a group, and let a,b <¢G be elements such that ab = ba. If the orders of a and b are
relatively prime, then prove that 0 (ab) =0 (a) 0 (6)

Show that any subring of a field is an integral domain.

Let G be an abelian group, and let n be any positive integer. Show that the function $:G, > Gy
defined by $(x)=<«” is ahomomorphism.

State and prove Fundamental Homomorphism Theorem.

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