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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.

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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.

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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.

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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>.

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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.

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

1. xⁿ
2. x_n
3. x^n-1
4. n^x

Answer: A) xⁿ

Explanation:

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

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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.

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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.

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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.

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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.

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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.

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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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