B Sc MATHEMATICS ABSTRACT ALGEBRA

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1 UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc MATHEMATICS (0 Admission Onwards) V Semester Core Course ABSTRACT ALGEBRA QUESTION BANK () Which of the following defines a binary operation on Z (a) (b) a b a b a b c, where c is the smallest integer greater than both a and b. (c) (d) a b c, where c is at least more than a b. a b c, where c is the largest integer less than the product of a and b. () Let Z be a set of integers, then under ordinary multiplication (Z,*) is (a) Group (b) Semi group (c) Monoid () If b and c are the inverses of some element a in a group G then : (a) b c (b) b c (c) b kc for some k N Abstract Algebra

2 (). How many non-trivial proper subgroups does the Klein -group have (a) 0 (b) (c) (d) () Under which of the following binary operation, the set of positive integers is closed (a) a b a ( b) a b a b b (c) a b a b ( d) a b ab () On Q, which of the following does not define a binary operation ( a) a b a b (b) a b a b (c) a b ab () Let be defined on Q as a b ab. Then identity for this binary operation is: (a) ( b) (c) 0 () The order of the subgroup of Z generated by the element is : ( a) (b) (c) (d) (9) Which of the following is a group (a) All n n diagonal matrices under addition (b) All n n diagonal matrices under matrix multiplication (c) All n n upper triangular matrices under matrix multiplication (0) Let G be a group and let a b c e for a, b, c G. Then b c a equals : (a) a ( b) e (c) b () Let be the binary operation defined on Q (a) a Then inverse of the element a is: (b) (c) a Abstract Algebra as a b ab. a () Value of the product ( ) ( - ) in Z is : (a) - (b) (c) 0 () Which of the following set addition is not a binary operation (a) Complex numbers (b) Real numbers (c) Non zero real numbers (d) Integers () How many binary operations can be defined on a set S with exactly n elements (a) (b) (c) (d)

3 () How many commutative binary operations can be defined on a set S with exactly n elements n( n) (a) n (b) ) (c) (d) () Which of the following are true () ( ) is a group. () is a group. () (Z, +) is a group. () (Z,. ) is a group. (a) & (b) Only (c) & (d) All () Which of the following are true () A group may have more than one identity element () Any two groups of three elements are isomorphic () Every group of at most three elements is abelian (a) & (b) & (c) & (d) All n n( n) () Let G= {, -, i, -i} where, be a set of four elements. Which of the following is a binary operation on G () a b a b () a b a.b (a) Only (b) Only (c) Both (9) Suppose a group G has an element x such that ax x for all a, then the number of elements in G is (a) (b) (c) (d) (0) Let G= C* ( non zero complex numbers). Which of the following are subgroups of G () All purely imaginary complex numbers under multiplication. () All complex numbers with absolute value under multiplication. (a) Only (b) Both (c) Only () Which of the following are true () ( ) is a subgroup of ( ). () is a subgroup of ( ) () ( ) is a subgroup of (R, +) (a) Only (b) Only (c) Only (d) All n n n () In a group G, a b a b for all a, b G. This statement is: (a) Always true (c) True if G is a multiplicative group (b) True if G is finite (d) True if G is abelian Abstract Algebra

4 () Which of the following is not true ( a) Z, is a proper subgroup of R, ( b) Q, is a proper subgroup of R,. (c) (d) Z, is a proper subgroup of Z, Q, is a proper subgroup of R, () Let k be a fixed positive integer and let kz kn n Z kz is a group which is isomorphic to R, /. Then : (a), (b) kz is not a group with respect to addition (c) kz, is a group which is isomorphic to Z, () The Klein - group is isomorphic to (a) Z Z (b) Z Z (c) Z () If G is a group of order four. Which of the following are not true () There exist a such that () There exist a such that () G is abelian (a) Only (b) & (c) Only (d) All () Consider with the binary operation m n = m + n. Which of the following are true () is commutative () is associative (a) Only (b) Only (c)all () Let (G, ) is a group. Which of the following are not always true () ( () = for all a, b in G () for all a, b in G (a) Only (b) Only (c) Only (d) & (9) Which of the following is a non-abelian group with the property that each proper subgroup is abelian (a) S (b) Z (c) S (0) The center of an abelian group G is: a) {e} b) G c) A cyclic subgroup d) None of these Abstract Algebra

5 () Let G be a cyclic group of order. Then the number of elements g G such that G = < g > is : ( a) (b) (c) (d) () Which of the following is true (a) Every cyclic group has a unique generator (b) In a cyclic group, every element is a generator (c) Every cyclic group has at least two generators () If a, b are elements of a group G of order m then order of ab and ba are: (a) same (b) Equal to m (c) unequal () Let G be a cyclic group of order. Then the number of elements g G such that G = < g > is : (a) (b) (c) (d) () How many elements are there on the cyclic subgroup of Z0 generated by (a) ( b) ( c) () Number of elements in the cyclic subgroup of the group of non-zero complex i numbers under multiplication, generated by, is: ( a) (b) (c) () Which of the following is a cyclic group with only one generator (a) Z (b) ( Z, ) ( c) Klein- group () How many generators are there for an infinite cyclic group (a) (b) (c) (d) Infinitely many (9) Which of the following is an example for a cyclic group with four generators (a) Z (b) Z (c) Z 9 (d) Z, (0) The least common multiple of r, s Z is rs if and only if: (a) r, s are relatively prime (b) One of them is a prime (c) Both are prime Abstract Algebra

6 () Which of the following is not true (a) Every cyclic group of order > has at least two distinct generators (b) If every proper subgroup of G is cyclic, then G itself is cyclic (c) Every sub group of a cyclic group is cyclic () Which of the following is a true statement (a) Every abelian group is cyclic (b) All generators of Z 0 are prime numbers (c) If G,G are groups, then G G is a group () How many elements are there in the subgroup generated by,, of the group Z (a) (b) (c) ( d) None of these () If, the order of the cyclic subgroup generated by is: (a) (b) (c) () Let. Then 00 (a) equals: (b) (c) () Let, then the orbit of under this permutation is: (a),,, (b),,,, (c),,, () Which of the following is a permutation function on R x (a) f ( x) x (b) f ( x) e (c) f ( x) x x x (d) f ( x) x (). Which of the following is true (a) Every function is a permutation if and only if it is one to one. (b) The symmetric group S is cyclic (c) The symmetric group S n is not cyclic for any n. (d) Every function from a finite set onto itself must be one to one. Abstract Algebra

7 Abstract Algebra (9) How many orbits are there for the permutation Z Z :, defined by ) ( n n (a) (b) (c) (d) infinitely many (0). How many orbits are there for the permutation Z Z :, defined by ) ( n n (a) (b) (c) (d) infinitely many (). How many elements are there in the orbit of under the permutation (a) (b) (c) (d) () The product,,,,, in S equals: (a) (b) (c) (d) () Which of the following represents (a),,,, (b),,,,, (c),,,, () Which of the following is an even permutation (a) (b) (c) (d) () Which of the following is not true (a) Every cycle is a permutation (b) A is a commutative group (c) A has 0 elements. () The order of (, 0 ) in Z Z is: (a) 9 ( b) (c)

8 () The order of (, 0, 9 ) in Z Z Z is: ( a) 0 (b) 0 (c) 0 () What is the largest possible order of a cyclic subgroup of Z Z (a) 0 (b) 0 (c) 0 (9) The cyclic subgroup of Z generated by has order (a) (b) ( c) 9 (0) The element (, ) of Z Z has order (a) (b) (c) (). Suppose G is a group of order 9. Then which of the following is true (a) G is not abelian (b) G has no subgroup other than (e) and G (c) There is a group H of order 9 which is not isomorphic to G (d) G is a subgroup of a group of order 0 () In a non-abelian group the element a has order 0. Then the order of a is: (a) (b) (c) (d) 9 () Let G be a group and g be fixed element of G. Then the map i g i g x gxg x G is : defined by (a) a one-one map on G, but not onto (b) an isomorphism of G with a proper subgroup of G (c) a bijective map from G to G, but not satisfies the homomorphism property (d) an isomorphism of G with itself. () If G is a group with no proper nontrivial subgroups, then G is: ( a) Not abelian (b) Cyclic (c) Isomorphic to Z, (d) An infinite group () How many automorphisms are there on the group Z (a) (b) (c) (d) None () How many subgroups are there for the group Z (a) (b) (c) () Which of the following group has no proper non trivial subgroups (a) Z ( b) Z (c) Z, (d) Klein- group Abstract Algebra

9 () If, then the smallest subgroup containing (a) (b) (c) (d) has order: (9) How many cosets are there of the subgroup Z of Z (a) (b) (c) (d) (0) Which of the following is not a coset of the subgroup of Z (a),,9 (b),,0 (c) 0,, ( d) None of these () The index of the subgroup in Z (a) (b) (c) (d) () If,,,,, then the index of in S (a) 0 (b) 0 ( c) 0 () If,,,,, then the index of in S (a) 0 (b) (c) () Which of the following is not true (a) Every subgroup of every group has left cosets (b) A subgroup of a group is a left coset of itself (c) A is of index in S for n > n n () Let G be a group having elements a and b such that O( a), O( b) and a b ba. Then O(ab) equals: ( a) (b) (c) (d) () The number of elements of order in the group Z, is: (a) (b) (c) (d) () Let G be any group and G G g g : ; ( ), g G (a) is a homomorphism (b) is a homomorphism if g is a finite group (c) is a homomorphism if g is abelian (d) is a homomorphism if and only if g is cyclic. Then: Abstract Algebra 9

10 (). Let : Z Z be a homomorphism such that ( ). Then Ker( ) equals: (a) Z (b) Z (c) Z (9) What is the value of (), where : Z Z be a homomorphism such that ( ) (a) 0 (b) (c) (d) (0) How many homomorphisms are there of Z onto Z (a) (b) (c) None (d) Infinitely many () Which of the following is true (a) Every cyclic group has prime order (b) Every abelian group is cyclic (c) Every group of prime order is cyclic () The sign of an even permutation is + and that of an odd permutation is -. Let sgn n : S n, be the homomorphism of Sn onto the multiplicative group defined by s gn n ( ) = sign of. Then kernel of is: (a) S n (b) An (c) { Identity permutation } () How many elements are there in the factor group Z / (a) ( b) ( c) (d) () Which of the following is true (a) Every homomorphism is a one to one map ( b) A homomorphism may have an empty kernel (c) For any two groups G and K, there exists a homomorphism of G into K (d) For any two groups G and K, there exists an isomorphism of G onto K () How many elements of finite order are there in Z Z Z (a) ( b) ( c) (d) Infinitely many () Let U be the multiplicative group of complex numbers of modulus and let xi : R, be the homomorphism defined as ( x) e, x R. Then U the kernel of is: (a) R (b) Z (c) Z (d) Abstract Algebra 0

11 () If G is an infinite cyclic group, then how many generators are there for the group G (a) Exactly two (b) At least two (c) Infinitely many (d) Only one () A cyclic group with only one generator can have at most elements (a) (b) (c) (d) infinitely many (9) The number of generators of the cyclic group of order is. (a) (b) (c) (d) (90) Which of the following is a homomorphism of groups (a) : R Z ; ( x) the greatest integer x. (b) : R R ; ( x) x under addition (c) Z Z ; ( ) the remainder of x when divided by (d) : 9 x : R R ; ( x ) x under multiplication (9) How many units are there in the ring Z Q Z (a) (b) (c) Infinitely many (d) none of these (9) The order of the ring M Z is : (a) (b) (c) ( d) None of these (9) A solution for the quadratic equation x x 0 in the ring Z is: (a) (b) (c) (d) (9). How many units are there in the ring Z Z (a) (b) (c) (9) Which of the following is not true (a) Every element in a ring has an additive inverse ( b) Multiplication in a field is commutative (c) Addition in every ring is commutative (d) Every ring with unity has at most two units (9) A solution for the quadratic equation x x 0 in the ring Z (a) - (b) (c) (d) is: (9) Which of the following ring have a non zero characteristic ( a) Z Z (b) Z Z (c) Z ( d) None of these Abstract Algebra

12 (9) What is the characteristic of Z Z (a) (b) (c) (99) Which of the following is true (a) The characteristic of nz is n. (b) nz has zero divisors if n is not a prime (c) As a ring, Z is isomorphic to nz; n (00) The solution for the equation x = in the field Z is: (a) (b) (c) (d) (0) The characteristic of the ring Z Z is: (a) (b) 0 (c) 90 (0) Let V be a vector space of dimension n. Which of the following is true (a) Any set containing n vectors is a basis for V (b) Any set containing n vectors linearly independent in V (c) Any set containing n vectors spans V (0) Which of the following sets of vectors form a basis of () {(0,,),(,0,),(0,,)} (){(,0,0),(0,0,),(0,,0)} (){(-,,),(,-,),(,-,)} (a) (b) (c) (d)all (0) Which of the following sets are vector spaces () Set of polynomials over R () Set of continuous functions on R ()Set of differentiable functions on R a)only & (b)only (c)all (d)none (0) The coordinate of (,,) relative to the ordered basis {(,0,)(,,0)(0,,)} of is (a) (,,) (b) (,,0) (c) (,,) (d)none (0) Let V be a vector space of all polynomials of degree n. Then dimension of V is (a)n (b)n- (c)n+ (d) (0) Let V be a vector space of dimension n. Which are true () Set of n+ vectors are linearly dependent ()V has only one basis ()Any set of n vectors is linearly independent (a)only (b) Only (c)only (d)all Abstract Algebra

13 (0) What is the dimension of the vector space of all matrices over R (a) (b) (c) (d) (b) (09) Which of the following is a subspace of (a) (, y, z) x y (c) (, y, z) x y z 0 R x (b) ( x, y, z) sin x 0 x (d) ( x, y, z) x y (0) Which of the following are true () is a ring () is a commutative ring () is a field () is a vector space over (a) & (b) Only (c) Only (d)all () How many ring homomorphisms are there from Z to Z (a) (b) (c) (d) () What is the dimension of the subspace W of, where W = {(a,b,c) : a=b=c } (a) (b) (c) (d) () Which of the following is a basis of the subspace W of, where W = {(a,b,c) : a=b=c } (a) {(,,)} (b) {(,,),(,,)} (c) {(,,)} () What is the co-ordinate vector of with respect to the ordered basis {, x, } (a) (0,,) (b)(0,0,) (c) (0,,0) (d)(,,) () Let W and W be subspaces of a vector space V. Then the smallest subspace of V containing both W and W is: (a) W (b). W (c) W W W W () Pick the incorrect statement: (a) If S spans V and S T, then T spans V (b) If T is linearly independent in V and S T, then S is linearly independent. (c) Any set of vectors that includes the zero vector is linearly dependent (d) Any single vector is linearly independent () Which of the following is a zero divisor in the ring (a) (b) (c) (d) 9 Abstract Algebra

14 () Value of m such that (m,, -) is a linear combination of the vectors (-,,) & (,,-) (a) (b) (c) (d) (9) The dimension of the subspace U = {(x,y,z) : x-y+z = 0} of is: (a) (b) (c) (b) (0) The coordinates of relative to the ordered basis (a) (,, ) (b) (,, ) (c) (,, ) (d) (,, ) Abstract Algebra

15 ANSWER KEY. b. c. a. d.c. c.b.b 9.a 0. b.b.b. c. b.d.b.a.b 9.a 0.c.b.d.d.c.b.a. d.b 9.a 0.b.c.d.a.d.b.a.a.b 9.a 0.a.b.d.b.a.c.a.d.d 9.b 0.a.b.d.a.d.c.a.c.a 9.a 0.a. b.d.d.b.a.a.b.b 9.a 0.d.a.c.c.d.d.a.c.c 9.b 0.a.c.b.a.c.c.b.a.b 9.d 90.d 9.c 9.c 9.d 9.b 9.d 9.c 9.d 9.a 99.d 00.c 0.b 0.d 0.b 0.c 0.a 0.c 0.a 0.b 09.c 0.a.b.a.a.a.c.d.b.b 9.b 0.a Abstract Algebra