Groups MCQ

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• An isomorphism of Boolean algebra is defined as _______
  ক

  order isomorphism

  খ

  unordered isomorphism

  গ

  order homomorphism

  ঘ

  hyper-morphism
• Invariant permutations of two functions can form __________
  ক

  groups

  খ

  lattices

  গ

  graphs

  ঘ

  rings
• How many indistinguishable necklaces can be made from beads of 4 colors with exactly 9 beads of each color where each necklace is of length 16?
  ক

  76967234

  খ

  5652209

  গ

  14414400

  ঘ

  8686214
• Let H be a finite group. The order of Sylow p-subgroup of H for every prime factor p with multiplicity 9 is?
  ক

  p+6

  খ

  p9

  গ

  pp

  ঘ

  3!*p2
• Suppose that H be an X-set and suppose that a∼b and |Xa|=|Xb|, the which of the following is true?
  ক

  Xa is powerset of Xb

  খ

  Xa is isomorphic to Xb

  গ

  Xa is homomorphic to Xb

  ঘ

  Xb is the subset of Xa
• _______ characterizes the properties of distributive lattices.
  ক

  Congruence Extension Property

  খ

  Algebraic extension property

  গ

  Poset

  ঘ

  Semigroup
• Which of the following is not an abelian group?
  ক

  semigroup

  খ

  dihedral group

  গ

  trihedral group

  ঘ

  polynomial group
• A set of representatives of all the cosets is called _________
  ক

  transitive

  খ

  reversal

  গ

  equivalent

  ঘ

  transversal
• The elements of a vector space form a/an ____________ under vector addition.
  ক

  abelian group

  খ

  commutative group

  গ

  associative group

  ঘ

  semigroup
• An isomorphism of a group onto itself is called ____________
  ক

  homomorphism

  খ

  heteromorphism

  গ

  epimorphism

  ঘ

  automorphism
• A function is defined by f(x)=2x and f(x + y) = f(x) + f(y) is called _____________
  ক

  isomorphic

  খ

  homomorphic

  গ

  cyclic group

  ঘ

  heteromorphic
• Lagrange’s theorem specifies __________
  ক

  the order of semigroup is finite

  খ

  the order of the subgroup divides the order of the finite group

  গ

  the order of an abelian group is infinite

  ঘ

  the order of the semigroup is added to the order of the group
• a * H = H * a relation holds if __________
  ক

  H is semigroup of an abelian group

  খ

  H is monoid of a group

  গ

  H is a cyclic group

  ঘ

  H is subgroup of an abelian group
• a * H is a set of _____ coset.
  ক

  right

  খ

  left

  গ

  sub

  ঘ

  semi
• Two groups are isomorphic if and only if __________ is existed between them.
  ক

  homomorphism

  খ

  endomorphism

  গ

  isomorphism

  ঘ

  association
• A normal subgroup is ____________
  ক

  a subgroup under multiplication by the elements of the group

  খ

  an invariant under closure by the elements of that group

  গ

  a monoid with same number of elements of the original group

  ঘ

  an invariant equipped with conjugation by the elements of original group
• What is a circle group?
  ক

  a subgroup complex numbers having magnitude 1 of the group of nonzero complex elements

  খ

  a subgroup rational numbers having magnitude 2 of the group of real elements

  গ

  a subgroup irrational numbers having magnitude 2 of the group of nonzero complex elements

  ঘ

  a subgroup complex numbers having magnitude 1 of the group of whole numbers
• The group of matrices with determinant _________ is a subgroup of the group of invertible matrices under multiplication.
  ক

  2

  খ

  3

  গ

  1

  ঘ

  4
• Intersection of subgroups is a ___________
  ক

  group

  খ

  subgroup

  গ

  semigroup

  ঘ

  cyclic group
• A group of rational numbers is an example of __________
  ক

  a subgroup of a group of integers

  খ

  a subgroup of a group of real numbers

  গ

  a subgroup of a group of irrational numbers

  ঘ

  a subgroup of a group of complex numbers
• __________ is not necessarily a property of a Group.
  ক

  Commutativity

  খ

  Existence of inverse for every element

  গ

  Existence of Identity

  ঘ

  Associativity
• Let K be a group with 8 elements. Let H be a subgroup of K and H<K. It is known that the size of H is at least 3. The size of H is __________
  ক

  8

  খ

  2

  গ

  3

  ঘ

  4
• Minimum subgroup of a group is called _____________
  ক

  a commutative subgroup

  খ

  a lattice

  গ

  a trivial group

  ঘ

  a monoid
• A trivial subgroup consists of ___________
  ক

  Identity element

  খ

  Coset

  গ

  Inverse element

  ঘ

  Ring
• An element a in a monoid is called an idempotent if ______________
  ক

  a-1=a*a-1

  খ

  a*a2=a

  গ

  a2=a*a=a

  ঘ

  a3=a*a
• A semigroup S under binary operation * that has an identity is called __________
  ক

  multiplicative identity

  খ

  monoid

  গ

  subgroup

  ঘ

  homomorphism
• ___________ are the symmetry groups used in the Standard model.
  ক

  lie groups

  খ

  subgroups

  গ

  cyclic groups

  ঘ

  poincare groups
• If the sum of elements in each row of an n×n matrix Z is zero, then the matrix is ______________
  ক

  inverse

  খ

  non-singular

  গ

  additive inverse

  ঘ

  singular
• If A, B, and C are invertible matrices, the expression (AB-1)-1(CA-1)-1C2 evaluates to ____________
  ক

  BC

  খ

  C-1BC

  গ

  AB-1

  ঘ

  C-1B
• If X is an idempotent nonsingular matrix, then X must be ___________
  ক

  singular matrix

  খ

  identity matrix

  গ

  idempotent matrix

  ঘ

  nonsingular matrix