# 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

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

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

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

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'''

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

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

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'

Explanation:

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

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