Discrete Mathematics | Subgroup MCQs

Discrete Mathematics | Subgroup MCQs: This section contains multiple-choice questions and answers on Subgroup in Discrete Mathematics.
Submitted by Anushree Goswami, on October 29, 2022

1. H is a subgroup of G if it is a _____ of G that is itself a group under G's operation.

  1. Void Set
  2. Non-void set
  3. Void Subset
  4. Non-void Subset

Answer: D) Non-void Subset

Explanation:

H is a subgroup of G if it is a non-void subset of G that is itself a group under G's operation.


2. Subsets of groups G are subgroups of G if:

  1. An identity element is a∈ H.
  2. The operation of G closes H, meaning that if a, b∈ H, then a, b∈ H
  3. Inverses of H have closed forms, i.e., if a∈ H then a-1∈ H.
  4. All of the above

Answer: B) The operation of G closes H, meaning that if a, b∈ H, then a, b∈ H

Explanation:

Subsets of groups G are subgroups of G if:

  1. An identity element is a∈ H.
  2. The operation of G closes H, meaning that if a, b∈ H, then a, b∈ H
  3. Inverses of H have closed forms, i.e., if a∈ H then a-1∈ H.

3. Subgroups K of a group G are said to be ____ subgroups if every element of K can be expressed in the form xn for some n ∈Z.

  1. Oval
  2. Cyclic
  3. Spherical
  4. Centric

Answer: B) Cyclic

Explanation:

Subgroups K of a group G are said to be cyclic subgroups if every element of K can be expressed in the form xn for some n ∈Z.


4. x is the _____ of group G's subgroup K, and K= <x>.

  1. Function
  2. Query
  3. Generator
  4. Supergroup

Answer: C) Generator

Explanation:

x is the generator of group G's subgroup K, and K= <x>.


5. We say that G is cyclic if ____, and x is its generator.

  1. G ≠ x
  2. G = x
  3. G != x
  4. G == x

Answer: B) G = x

Explanation:

We say that G is cyclic if G = x, and x is its generator.


6. A group G is cyclic if every element of G can be written as __ for some n∈ Z.

  1. xn
  2. xn
  3. xn-1
  4. nx

Answer: A) xn

Explanation:

A group G is cyclic if every element of G can be written as xn for some n∈ Z.


7. Suppose we have an algebraic system (G,*), where * is a binary operation on G. An abelian group is one which satisfies all of the group's properties plus the ____ property of the group's operation.

  1. Closed
  2. Associative
  3. Identity
  4. All of the above

Answer: C) Identity

Explanation:

Suppose we have an algebraic system (G,*), where * is a binary operation on G. An abelian group is one which satisfies all of the group's properties plus the closed, associative, identity, inverse and commutative property of the group's operation.


8. Let G be a group and H a subgroup. It is possible to express the elements of a left coset of H in G as xH={ __ | h ∈ H } for any x∈ G.

  1. xh
  2. hx
  3. xxh
  4. hhx

Answer: A) xh

Explanation:

Let G be a group and H a subgroup. It is possible to express the elements of a left coset of H in G as xH={ xh | h ∈ H } for any x∈ G.


9. A right coset of H in G is a subset with radius Hx = {__ | h ∈H }, for any x∈G.

  1. xh
  2. hx
  3. xxh
  4. hhx

Answer: B) hx

Explanation:

A right coset of H in G is a subset with radius Hx = {hx | h ∈H }, for any x∈G.


10. A ____ coset and a ____ coset are respectively called complexes xH and Hx.

  1. Left, right
  2. Right, Left
  3. Left, left
  4. Right, right

Answer: A) Left, right

Explanation:

A left coset and a right coset are respectively called complexes xH and Hx.


11. If the group operation is additive (+), then _____ = {x+h | h ∈H} denotes a left coset.

  1. x + H
  2. H + x
  3. Both A and B
  4. None of the above

Answer: A) x + H

Explanation:

If the group operation is additive (+), then x + H = {x+h | h ∈H} denotes a left coset.


12. If the group operation is additive (+), then _____ = {x+h | h ∈H} denotes a right coset.

  1. x + H
  2. H + x
  3. x - H
  4. H - x

Answer: B) H + x

Explanation:

If the group operation is additive (+), then H + x = {h+x | h ∈H} denotes a left coset.






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