Discrete Mathematics | Normal Subgroup MCQs

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

1. It is a normal subgroup of G if for all h∈ H and x∈ G, ____∈ H.

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

Answer: A) x h x-1

Explanation:

It is a normal subgroup of G if for all h∈ H and x∈ G, x h x-1∈ H.


2. When x H x-1 = [x h x-1| h ∈ H} then H is normal in G ____ x H x-1⊆H, ∀ x∈ G.

  1. If
  2. If and only if
  3. If not
  4. None of the above

Answer: B) If and only if

Explanation:

When x H x-1 = [x h x-1| h ∈ H} then H is normal in G if and only if x H x-1⊆H, ∀ x∈ G.


3. The subgroup H of an abelian group G is normal in G if G is an ____ group.

  1. Abelian
  2. Normal
  3. Sub
  4. None of the above

Answer: A) Abelian

Explanation:

The subgroup H of an abelian group G is normal in G if G is an abelian group.


4. Homomorphisms are mappings such that ____, x, y ∈ G.

  1. f (xy) =f(x) f(y)
  2. f (xy) =f(x) + f(y)
  3. f (xy) =f(x) - f(y)
  4. f (xy) =f(x) / f(y)

Answer: A) f (xy) =f(x) f(y)

Explanation:

Homomorphisms are mappings such that f (xy) =f(x) f(y), x, y ∈ G.


5. Even though the binary operations of the groups G and G' are different, the mapping f preserves the ____ operation.

  1. Group
  2. Subgroup
  3. Supergroup
  4. None

Answer: A) Group

Explanation:

Even though the binary operations of the groups G and G' are different, the mapping f preserves the group operation.


6. Even though the binary operations of the groups G and G' are different, the mapping f preserves the group operation. This condition is known as -

  1. Hypermorphism
  2. Homomorphism
  3. Heteromorphism
  4. Hypomorphism

Answer: B) Homomorphism

Explanation:

Even though the binary operations of the groups G and G' are different, the mapping f preserves the group operation. This condition is known as homomorphism.


7. A homomorphism of a group G to a group G' with identity e' is a homomorphism with a kernel {x∈ G | f(x) =__'}.

  1. e
  2. e'
  3. e''
  4. e'''

Answer: B) e'

Explanation:

A homomorphism of a group G to a group G' with identity e' is a homomorphism with a kernel {x∈ G | f(x) =e'}


8. ____ f represents the kernel of f.

  1. f
  2. K f
  3. Ker f
  4. None

Answer: C) Ker f

Explanation:

Ker f represents the kernel of f.


9. The ____ set of f consists of the range of the map f, denoted by f (G).

  1. Direction
  2. Line
  3. Image
  4. Circle

Answer: C) Image

Explanation:

The image set of f consists of the range of the map f, denoted by f (G).


10. Homomorphic images of G are those whose f (G) = ____.

  1. G
  2. G'
  3. F
  4. F'

Answer: B) G'

Explanation:

Homomorphic images of G are those whose f (G) = G'.





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