B.Sc. Part 1 Mathematics
ALGEBRA
Ques1: If N is the set of natural numbers, then prove that the relation R defined on N x N by (a,b)R(c,d) if and only if ad = bc , is an equivalence relation. [Kanpur B.Sc. 2013]
Ques2: If A = {1,2,3,4} and R = { (1,1), (2,2), (2,3), (3,2), (3,3), (4,4) } be an equivalence relation on A, then find the partition of A induced by R. [Avadh 2012]
Ques3: If S = {1,2,3,4,5} and C = { (1,2), (3), (4,5) }, then prove that the partition C defines an equivalence relation on S. [Rohil. 2011]
Ques4: If G = {0} be a singleton set with 0 (zero) as its only element, show that it forms a group with respect to addition operation. [Agra 2007]
Ques5: The set Z of integers forms an infinite abelian group with respect to the operation of addition, i.e., (Z,+) is an infinite abelian group. [Meerut 2013; Rohil. 2010; Bundel. 2009]
Ques6: Write the composition table for the elements in the set of cube roots of unity. [Kashi 2012]
Ques7: Four fourth roots of unity form a finite abelian group with respect to multiplication. [Kanpur B.Sc. 1993,2014; Rohil. 2010,12; Meerut 2000; Purvanchal 2012,14; Gorakh. 2006; Agra 2008; Bundel. 2001,04,10]
Ques8: The n th roots of unity form an abelian group under multiplication. [Kanpur B.Sc. 1996; Avadh 2007; Rohil. 2006; Gorakh. 2007]
Ques9: The set of all n x n non-singular matrices over the set Q (or R or C) is a non-abelian group under the operation of multiplication. [Meerut 1996]
Ques10: The set of residue classes modulo 5 is an abelian group with respect to the addition of residue classes modulo 5. [Bundel. 2008; Kanpur B.Sc. 2013]
Ques11: Prove that the set G = {0,1,2,3,4,5} is an abelian group of order 6 w.r.t. addition modulo 6. [Bundel. 2014; Meerut 2014]
Ques12: Give an example of a group of order 2. [Lucknow 2008]
Ans: The set G = { [0], [1] } of residue classes modulo 2 under the addition of residue classes.
Ques13: The set of residue classes modulo m is a finite abelian group with respect to the addition of residue classes modulo m. [Kanpur B.Sc. 2000]
Ques14: The set { [1], [2], [3], [4], [5], [6] } of non-zero residue classes modulo 7 forms an abelian group under the multiplication of residue classes modulo 7. [Kanpur B.Sc. 2004, 08; Meerut 1993]
Ques15: The set of non-zero residue classes modulo p, where p is prime, is a finite abelian group with respect to the multiplication of residue classes modulo p. [Avadh 2008; Rohil. 2007, 14]
Ques16: Does the set G = {1,2,3} form a group with respect to multiplication modulo 4 ? [Avadh 2012]
Ans: No; since [2].[2] = [4] = [0] is not the element of G.
Ques17: For the binary operation * defined by x * y = x + y - 5, obtain the identity if it exists. [Lucknow 2004]
Ans: We get that
x * e = x
x + e - 5 = x
e = 5.
Hence 5 is the required identity.
Ques18: Show that if every element of a group G is its own inverse, then G is abelian. [Kanpur B.Sc. 1996,2004; Bundel. 2005,06,10; Purvanchal 2008,13; Avadh 2013; Lucknow 2006; Agra 2009]
Ques19: Prove that a group with three elements is necessarily abelian. [Purvanchal 2011]
Ques20: Define a group and give an example of a finite non-abelian group. [Kanpur 2005, 09]
Ques21: Check : (1) the set of all even permutations on n objects, (2) the set of all odd permutations on n objects, for being a group under permutation multiplications. [Lucknow 2006]
Ans: (1) It is a group.
(2) It is not a group.
Ques22: Define binary operation and state group axioms. [Kanpur B.Sc.1994, 2001; Meerut 1994]
Ques23: Define group. Show that the set of all odd integers with addition operation is not a group. [Agra 2014]
Ques24: Define a group. Show that set Q+ of all positive rational numbers forms an abelian group with respect to multiplication of numbers. [Bundel. 2003]
Ques25: Define a semigroup. Give an example also. [Kanpur B.Sc. 2001; Avadh 2016]
Ques26: Define an abelian group. Show that the set of non-zero complex numbers of the form a+ib forms an infinite abelian group with respect to multiplication. [Meerut 1995]
Ques27: Prove that every group of order 3 is abelian. [Kanpur B.Sc. 2010]
Ques28: Show that the set G = {5,1/5,1,-1} is not a group with respect to multiplication operation. [Bundel. 2014]
Ques29: Give an example for each of the following :
1) A non-abelian groupoid.
2) A non-abelian semigroup.
3) A non-abelian monoid.
4) A non- abelian group. [Kanpur B.Sc. 1994]
Ques30: Prove that the set of all three dimensional vectors forms an infinite abelian group with respect to vector addition as composition. [Rohil. 2006; Meerut 1994]
Ques31: Write the order of each element of the multiplicative group {1,-1,i,-i}. [Kanpur B.Sc. 2003; Kashi 2012; Meerut 2005; Rohil. 2007]
Ques32: Let G = { [0], [1], [2], [3] } be the additive group of residue classes modulo 4. Find the order of each element. [Kanpur B.Sc. 2007; Agra 2014]
Ques33: Show that a group is abelian if its every element, except the identity, is of order 2. [Kanpur B.Sc. 2005]
Ques34: Find the order of each element of the additive group G = { [0], [1], [2], [3], [4] } of residue classes modulo 5. [Meerut 2010]
Ques35: Define subgroup of a group G and give an example of subgroup with respect to multiplication. [Kanpur B.Sc. 2007; Rohil. 2008; Meerut 1995, 2000; Avadh 2006; Gorakh. 2007; Agra 2014]
Ques36: Find the subgroups of the multiplicative group {1,i,-1,-i}. [Bundel. 2011]
Ques37: Give an example of abelian group whose all subgroups are abelian. [Kanpur B.Sc. 2014]
Ans: The multiplicative group G = {1,-1,i,-i} of fourth roots of unity.
Ques38: Find the number of generators of a cyclic group of order 16. [Lucknow 2007, 10]
Ans: 8
Ques39: Show that the residue classes [1], [2], [3], [4], [5], [6] mod 7 form a multiplicative cyclic group. Find the number of its generators. [Lucknow 2004; Meerut 1991]
Ques40: Find all the generators of a cyclic groups of order 7. [Kanpur B.Sc.2004]
Ques41: Prove that every group of order 3 is cyclic. [Kanpur B.Sc. 1998, 2009, 10, 12; Purvanchal 2010]
Ques42: Prove that the set of n th roots of unity forms a multiplicative cyclic group. [Rohil. 2006; Kashi 2011; Agra 2013]
Ques43: Give an example of abelian group which is not cyclic. [Kashi 2011]
Ques44: Define cyclic group. Give an example of non-cyclic group whose all proper subgroups are cyclic. [Kanpur B.Sc. 2005]
Ques45: Give an example of a group whose all subgroups are cyclic. [Kanpur B.Sc. 2014]
Ques46: Define a cyclic group and give an example. [Kanpur B.Sc. 1992,94,2004; Avadh 2009; Gorakh 2007; Agra 2000; Meerut 1995, 2003; Kumaun 2000; Garhwal 2001; Bundel. 2008]
Ques47: Prove that the multiplicative group {1.i,-1,-i} is cyclic. Find the generators of this group. [Kanpur B.Sc. 1990]
Ques48: Find the number of generators of a cyclic group of order 8. [Kanpur B.Sc. 2014; Lucknow 2009; Agra 2005]
Ques49: Let G be a cyclic group of order 10. What is the number of distinct generators of G. [Meerut 2005]
Ques50: Give an example of a finite abelian group which is not cyclic. [Bundel. 2009; Kashi 2011]
Ques51: Show that every finite group of order less than six must be abelian. [Avadh 2004; Kanpur B.Sc. 1994, 96]
Ques52: Obtain coset decomposition of the multiplicative group G = {1,i,-1,-i} with respect to its subgroup H = {1,-1}. [Rohil. 2011; Kashi 2012; Avadh 2012]
Ques53: Give an example of a non-abelian group whose all proper subgroups are cyclic. [Kanpur B.Sc. 2013]
Ques54: Let G be a group and H and K are finite subgroups of group G of respective order 9 and 20. The find the number of distinct elements in HK. [Kanpur B.Sc. 2005, 10]
Ques55: Define cosets. Illustrate by an example. [Lucknow 2004, 10]
Ques56: State Lagrange's theorem. [Avadh 2014]
Ques57: Define isomorphism and prove that every group of order 3 is isomorphic to every other group of group 3. [Kanpur B.Sc. 2001; Meerut 1991; Bundel. 2005, 14]
Ques58: Define homomorphism of groups and give its properties. Give two examples also. [Kanpur B.Sc. 2003, 07; Purvanchal 2014; Bundel. 2001, 14]
Ques59: Define homomorphism and isomorphism of groups. [Bundel. 2013]
Ques60: Every subgroup of an abelian group is normal. [Avadh 2005; Rohil. 2014; Bundel. 2012, 13; Meerut 2009,10]
Ques61: Prove that a group of prime order is a simple group. [Kanpur B.Sc. 2003]
Ques62: State and prove class equation. [Kanpur B.Sc. 2014]
Ques63: Define the centre of a group and prove that it is always a normal subgroup of the group. [Kanpur B.Sc. 1991, 92]
Ques64: Show that if H is the only subgroup of finite order of a group G, then H is normal in G. [Kanpur B.Sc. 1991]
Ques65: Prove that a subgroup N of G is a normal subgroup of G if and only if every left coset of N in G is a right coset of N in G. [Kanpur B.Sc. 1991]
Ques66: If f is a homomorphism of a group G onto G' with kernel K, then prove that G' is isomorphic to G/K. [Kanpur B.Sc. 1991]
Ques67: If a cyclic subgroup of a group N of G is normal in G, then show that every subgroup of N is normal in G. [Kanpur B.Sc.1992]
Ques68: If M is a subgroup of a group G and N is a normal subgroup of G, then prove that NM is a subgroup of G. [Kanpur B.A. 1991; Bundel. B.Sc. 1992]
Ques69: What do you mean by the relation of conjugacy in a group? Prove that conjugacy is an equivalence relation in a group. [Kanpur B.Sc. 1991]
Ques70: The algebraic structure (Z,+,.), i.e., the set of integers with ordinary addition and multiplication as compositions is a commutative ring with unity. [Bundel. 2006, 09; Purvanchal 2009, 11]
Ques71: Show that the ring of integers is an integral domain. [Avadh 2008]
Ques72: Prove that the set of all residue classes modulo prime integer p forms a finite field with respect to addition and multiplication of residue classes modulo p. [Bundel. 2007]
Ques73: Prove that the set S = {0,1} (mod 2) is a field with respect to ordinary addition and multiplication. [Kanpur B.Sc. 2004]
Ques74: Prove that the set {0,1,2} (mod 3) under addition and multiplication is a field. [Kanpur B.Sc. 2000]
Ques75: Prove that a skew field is without zero divisors. [Kanpur B.Sc. 2002; Meerut 2013]
Ques76: Define ring. [Avadh 2007; Bundel. 2005, 09, 10; Kanpur B.Sc. 2005]
Ques77: Define ring with unity. [Kanpur B.Sc. 2006]
Ques78: Define ring with proper zero divisors. [Kanpur B.Sc. 2002; Meerut 2003]
Ques79: Define commutative ring without zero divisors. [Lucknow 2008]
Ques80: Define integral domain. [Rohil. 2008; Lucknow 2006, 10; Purvanchal 2014; Kanpur B.Sc. 1997, 2001, 02, 03, 04, 05]
Ques81: Define field. [Kanpur B.Sc. 2004, 05; Rohil. 2008; Lucknow 2006]
Ques82: Give an example of ring but not an integral domain. [Lucknow 2006]
Ques83: Give an example of integral domain but not a field. [Lucknow 2006]
Ques84: Prove that the singleton set {0} is a ring with respect to ordinary addition and multiplication. [Purvanchal 2012]
Ques85: Prove that the set E of all even integers forms a commutative ring but not a field. [Bundel. 2014]
Ques86: Prove that the set of all n x n matrices over the field of real numbers is a ring with respect to matrix addition and multiplication. Is it a commutative ring ? Does this ring possess zero divisors ? [Lucknow 2016]
Ques87: Prove that the set E of even integers is a subring of the ring ( Z, +, . ) of integers. [Kanpur B.Sc. 2003; Avadh 2012]
Ques88: Give an example of a ring with identity whose subring is without identity. [Kanpur B.Sc. 2013]
Ques89: If f is a homomorphism of a ring R into R', show that for each subring S of R, f(S) is a subring of R'. [Kanpur B.A. 87, B.Sc. 1983]
Ques22: Define binary operation and state group axioms. [Kanpur B.Sc.1994, 2001; Meerut 1994]
Ques23: Define group. Show that the set of all odd integers with addition operation is not a group. [Agra 2014]
Ques24: Define a group. Show that set Q+ of all positive rational numbers forms an abelian group with respect to multiplication of numbers. [Bundel. 2003]
Ques25: Define a semigroup. Give an example also. [Kanpur B.Sc. 2001; Avadh 2016]
Ques26: Define an abelian group. Show that the set of non-zero complex numbers of the form a+ib forms an infinite abelian group with respect to multiplication. [Meerut 1995]
Ques27: Prove that every group of order 3 is abelian. [Kanpur B.Sc. 2010]
Ques28: Show that the set G = {5,1/5,1,-1} is not a group with respect to multiplication operation. [Bundel. 2014]
Ques29: Give an example for each of the following :
1) A non-abelian groupoid.
2) A non-abelian semigroup.
3) A non-abelian monoid.
4) A non- abelian group. [Kanpur B.Sc. 1994]
Ques30: Prove that the set of all three dimensional vectors forms an infinite abelian group with respect to vector addition as composition. [Rohil. 2006; Meerut 1994]
Ques31: Write the order of each element of the multiplicative group {1,-1,i,-i}. [Kanpur B.Sc. 2003; Kashi 2012; Meerut 2005; Rohil. 2007]
Ques32: Let G = { [0], [1], [2], [3] } be the additive group of residue classes modulo 4. Find the order of each element. [Kanpur B.Sc. 2007; Agra 2014]
Ques33: Show that a group is abelian if its every element, except the identity, is of order 2. [Kanpur B.Sc. 2005]
Ques34: Find the order of each element of the additive group G = { [0], [1], [2], [3], [4] } of residue classes modulo 5. [Meerut 2010]
Ques35: Define subgroup of a group G and give an example of subgroup with respect to multiplication. [Kanpur B.Sc. 2007; Rohil. 2008; Meerut 1995, 2000; Avadh 2006; Gorakh. 2007; Agra 2014]
Ques36: Find the subgroups of the multiplicative group {1,i,-1,-i}. [Bundel. 2011]
Ques37: Give an example of abelian group whose all subgroups are abelian. [Kanpur B.Sc. 2014]
Ans: The multiplicative group G = {1,-1,i,-i} of fourth roots of unity.
Ques38: Find the number of generators of a cyclic group of order 16. [Lucknow 2007, 10]
Ans: 8
Ques39: Show that the residue classes [1], [2], [3], [4], [5], [6] mod 7 form a multiplicative cyclic group. Find the number of its generators. [Lucknow 2004; Meerut 1991]
Ques40: Find all the generators of a cyclic groups of order 7. [Kanpur B.Sc.2004]
Ques41: Prove that every group of order 3 is cyclic. [Kanpur B.Sc. 1998, 2009, 10, 12; Purvanchal 2010]
Ques42: Prove that the set of n th roots of unity forms a multiplicative cyclic group. [Rohil. 2006; Kashi 2011; Agra 2013]
Ques43: Give an example of abelian group which is not cyclic. [Kashi 2011]
Ques44: Define cyclic group. Give an example of non-cyclic group whose all proper subgroups are cyclic. [Kanpur B.Sc. 2005]
Ques45: Give an example of a group whose all subgroups are cyclic. [Kanpur B.Sc. 2014]
Ques46: Define a cyclic group and give an example. [Kanpur B.Sc. 1992,94,2004; Avadh 2009; Gorakh 2007; Agra 2000; Meerut 1995, 2003; Kumaun 2000; Garhwal 2001; Bundel. 2008]
Ques47: Prove that the multiplicative group {1.i,-1,-i} is cyclic. Find the generators of this group. [Kanpur B.Sc. 1990]
Ques48: Find the number of generators of a cyclic group of order 8. [Kanpur B.Sc. 2014; Lucknow 2009; Agra 2005]
Ques49: Let G be a cyclic group of order 10. What is the number of distinct generators of G. [Meerut 2005]
Ques50: Give an example of a finite abelian group which is not cyclic. [Bundel. 2009; Kashi 2011]
Ques51: Show that every finite group of order less than six must be abelian. [Avadh 2004; Kanpur B.Sc. 1994, 96]
Ques52: Obtain coset decomposition of the multiplicative group G = {1,i,-1,-i} with respect to its subgroup H = {1,-1}. [Rohil. 2011; Kashi 2012; Avadh 2012]
Ques53: Give an example of a non-abelian group whose all proper subgroups are cyclic. [Kanpur B.Sc. 2013]
Ques54: Let G be a group and H and K are finite subgroups of group G of respective order 9 and 20. The find the number of distinct elements in HK. [Kanpur B.Sc. 2005, 10]
Ques55: Define cosets. Illustrate by an example. [Lucknow 2004, 10]
Ques56: State Lagrange's theorem. [Avadh 2014]
Ques57: Define isomorphism and prove that every group of order 3 is isomorphic to every other group of group 3. [Kanpur B.Sc. 2001; Meerut 1991; Bundel. 2005, 14]
Ques58: Define homomorphism of groups and give its properties. Give two examples also. [Kanpur B.Sc. 2003, 07; Purvanchal 2014; Bundel. 2001, 14]
Ques59: Define homomorphism and isomorphism of groups. [Bundel. 2013]
Ques60: Every subgroup of an abelian group is normal. [Avadh 2005; Rohil. 2014; Bundel. 2012, 13; Meerut 2009,10]
Ques61: Prove that a group of prime order is a simple group. [Kanpur B.Sc. 2003]
Ques62: State and prove class equation. [Kanpur B.Sc. 2014]
Ques63: Define the centre of a group and prove that it is always a normal subgroup of the group. [Kanpur B.Sc. 1991, 92]
Ques64: Show that if H is the only subgroup of finite order of a group G, then H is normal in G. [Kanpur B.Sc. 1991]
Ques65: Prove that a subgroup N of G is a normal subgroup of G if and only if every left coset of N in G is a right coset of N in G. [Kanpur B.Sc. 1991]
Ques66: If f is a homomorphism of a group G onto G' with kernel K, then prove that G' is isomorphic to G/K. [Kanpur B.Sc. 1991]
Ques67: If a cyclic subgroup of a group N of G is normal in G, then show that every subgroup of N is normal in G. [Kanpur B.Sc.1992]
Ques68: If M is a subgroup of a group G and N is a normal subgroup of G, then prove that NM is a subgroup of G. [Kanpur B.A. 1991; Bundel. B.Sc. 1992]
Ques69: What do you mean by the relation of conjugacy in a group? Prove that conjugacy is an equivalence relation in a group. [Kanpur B.Sc. 1991]
Ques70: The algebraic structure (Z,+,.), i.e., the set of integers with ordinary addition and multiplication as compositions is a commutative ring with unity. [Bundel. 2006, 09; Purvanchal 2009, 11]
Ques71: Show that the ring of integers is an integral domain. [Avadh 2008]
Ques72: Prove that the set of all residue classes modulo prime integer p forms a finite field with respect to addition and multiplication of residue classes modulo p. [Bundel. 2007]
Ques73: Prove that the set S = {0,1} (mod 2) is a field with respect to ordinary addition and multiplication. [Kanpur B.Sc. 2004]
Ques74: Prove that the set {0,1,2} (mod 3) under addition and multiplication is a field. [Kanpur B.Sc. 2000]
Ques75: Prove that a skew field is without zero divisors. [Kanpur B.Sc. 2002; Meerut 2013]
Ques76: Define ring. [Avadh 2007; Bundel. 2005, 09, 10; Kanpur B.Sc. 2005]
Ques77: Define ring with unity. [Kanpur B.Sc. 2006]
Ques78: Define ring with proper zero divisors. [Kanpur B.Sc. 2002; Meerut 2003]
Ques79: Define commutative ring without zero divisors. [Lucknow 2008]
Ques80: Define integral domain. [Rohil. 2008; Lucknow 2006, 10; Purvanchal 2014; Kanpur B.Sc. 1997, 2001, 02, 03, 04, 05]
Ques81: Define field. [Kanpur B.Sc. 2004, 05; Rohil. 2008; Lucknow 2006]
Ques82: Give an example of ring but not an integral domain. [Lucknow 2006]
Ques83: Give an example of integral domain but not a field. [Lucknow 2006]
Ques84: Prove that the singleton set {0} is a ring with respect to ordinary addition and multiplication. [Purvanchal 2012]
Ques85: Prove that the set E of all even integers forms a commutative ring but not a field. [Bundel. 2014]
Ques86: Prove that the set of all n x n matrices over the field of real numbers is a ring with respect to matrix addition and multiplication. Is it a commutative ring ? Does this ring possess zero divisors ? [Lucknow 2016]
Ques87: Prove that the set E of even integers is a subring of the ring ( Z, +, . ) of integers. [Kanpur B.Sc. 2003; Avadh 2012]
Ques88: Give an example of a ring with identity whose subring is without identity. [Kanpur B.Sc. 2013]
Ques89: If f is a homomorphism of a ring R into R', show that for each subring S of R, f(S) is a subring of R'. [Kanpur B.A. 87, B.Sc. 1983]
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