assignment mathematics algebra

CSE/IT/SEM-4/M-401/ABSTRACT ALGEBRA/ASSIGNMENT-2014

ASSIGNMENT-1/2014

GROUP - A

(Multiple Choice Type Questions)

1. Choose the correct alternatives for the following:

i) Let G be a Group and a; b 2 G. Then (a1b)1 is equal toa) ab1 b) b1a

c) a1b1 d) b1a1

ii) The set f[2]; [4]; [6]; [8]g is a group under multiplication modulo 10. The identity element ofthe group is

a) [2] b) [4]

c) [6] d) [8]

iii) Let G be a nite group of even order. Then the number of elements of order 2 in G is

a) 2 b) 4

c) even d) odd

iv) The generators of the cyclic group (Z;+) are

a) 1, -1 b) 0, 1

c) 0, -1 d) 2, -2

v) Let f : G! H be a group homomorphism. Then Im(f) is aa) Cyclic subgroup of G b) Cyclic subgroup of H

c) Normal subgroup of G d) Normal subgroup of H

vi) The number of elements of order 8 in a cyclic group of order 8 is

a) 1 b) 8

c) 2 d) 4

vii) In the group S3 of all permutations on f1; 2; 3g; the inverse of0@ 1 2 3

2 3 1

1A isa)

0@ 1 2 31 2 3

1A b)0@ 1 2 3

1 3 2

1Ac)

0@ 1 2 33 1 2

1A d)0@ 1 2 3

3 2 1

1A


viii) The number of unit elements of the ring (Z;+; ) isa) 2 b) 3

c) 1 d) innite

ix) Let Z be the set of all integers and be a binary operation on Z dened by ab = a+b+10for all a; b 2 Z. The identity element of the group (Z;) isa) 0 b) 10

c) -10 d) 1

x) Let Z be the set of all integers and be a binary operation on Z dened by ab = a+b+1for all a; b 2 Z. Then the inverse of the element 5 isa) 5 b) -5

c) -6 d) -7

xi) In an Abelian group (a) = 5 and (b) = 7. Then (ab)14 is equal toa) a b) a1

c) b d) ab

GROUP - B

(Short Answer Type Questions)

2. Prove that a group (G; ) is commutative if and only if (a b)2 = a2 b2, for all a; b 2 G.

3. Prove that the identity element of a group is unique.

4. Let (G; ) be a group and a 2 G. Prove that there exists an unique element b 2 G suchthat a b = b a = e, where `e0 is the identity element of G.

5. Let (G; ) be a group and a 2 G such that (a) = n.

(a) If am = e for some positive integer m, then prove that n divides m.

(b) Prove that for every positive integer t, (at) = ngcd(t;n) .

6. Consider all the complex roots of the equation xn = 1. Show that they form a group under

the usual complex multiplication.

7. Let U9 = f[a] 2 Z9 : 0 < a < 9 and gcd(a; 9) = 1g. Show that U9 forms a group undermultiplication modulo 9.

8. Find the order of [3] in U14.

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9. Let (G; ) be a group and a; b 2 G. Suppose that a2 = e and a b4 a = b7. Prove thatb33 = e.

10. In the group Z15 nd the order of the following elements: [5]; [8] and [10].

11. Let G be a group and a 2 G. If (a) = 24, then nd out (a4); (a7) and (a10).

12. Let G = fa 2 R : 1 < a < 1g. Dene a binary operation on G by a b = a+b1+ab for alla; b 2 G. Show that (G; ) is a group.

13. Find out the order of each element in the Group S3.

14. Find out the inverse element of (123) 2 S3.

15. Find out the order of the permutation (1234)(56) 2 S6.

16. Find out the inverse of the permutation (1234)(56) 2 S6.

17. Prove that intersection of any two subgroup of a group G is also the subgroup of G.

18. Give a counterexample to show that union of two subgroups of G is not necessarily a

subgroup of G.

19. Let G be a group and H be a nonempty nite subset of G. Then H is a subgroup of G if

and only if for all a; b 2 H; ab 2 H.

20. Let (G; ) be a group and a 2 G. Then the Centraliser of a is dened by C(a) = fx 2G : x a = a xg. Prove that C(a) is a subgroup of G.

21. Let (G; ) be a group. The Center of the group G is dened by Z(G) = fx 2 G : g x =x g for all g 2 Gg. Prove that Z(G) is a subgroup of G.

22. If G is a commutative group , then prove that H = fa 2 G : a2 = eg is a subgroup of G.

23. Show that every cyclic group is commutative. Give a counterexample to show that every

commutative group is not cyclic.

24. Show that the 7 th roots of unity form a cyclic group. Find all the generators of this

group.

25. Prove that every subgroup of a cyclic group is cyclic.

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26. Let H be a subgroup of a nite group G. Prove that the order of H divides the order of G.

27. Let H be a subgroup of a group G. Then prove that faH : a 2 Gg forms a partition of G.

28. Prove that every group of prime order is cyclic.

29. Let G be a group of nite order n and a 2 G. Then (a) divides n and an = e.

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assignment mathematics algebra

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