Discrete Mathematics Questions and Answers – Groups – Subgroups

«
»

This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Groups – Subgroups”.

1. A trivial subgroup consists of ___________
a) Identity element
b) Coset
c) Inverse element
d) Ring
View Answer

Answer: a
Explanation: Let G be a group under a binary operation * and a subset H of G is called a subgroup of G if H forms a group under the operation *. The trivial subgroup of any group is the subgroup consisting of only the Identity element.
advertisement

2. Minimum subgroup of a group is called _____________
a) a commutative subgroup
b) a lattice
c) a trivial group
d) a monoid
View Answer

Answer: c
Explanation: The subgroups of any given group form a complete lattice under inclusion termed as lattice of subgroups. If o is the Identity element of a group(G), then the trivial group(o) is the minimum subgroup of that group and G is the maximum subgroup.

3. 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 __________
a) 8
b) 2
c) 3
d) 4
View Answer

Answer: d
Explanation: For any finite group G, the order (number of elements) of every subgroup L of G divides the order of G. G has 8 elements. Factors of 8 are 1, 2, 4 and 8. Since given the size of L is at least 3(1 and 2 eliminated) and not equal to G(8 eliminated), the only size left is 4. Size of L is 4.

4. __________ is not necessarily a property of a Group.
a) Commutativity
b) Existence of inverse for every element
c) Existence of Identity
d) Associativity
View Answer

Answer: a
Explanation: Grupoid has closure property; semigroup has closure and associative; monoid has closure, associative and identity property; group has closure, associative, identity and inverse; the abelian group has group property and commutative.

5. A group of rational numbers is an example of __________
a) a subgroup of a group of integers
b) a subgroup of a group of real numbers
c) a subgroup of a group of irrational numbers
d) a subgroup of a group of complex numbers
View Answer

Answer: b
Explanation: If we consider the abelian group as a group rational numbers under binary operation + then it is an example of a subgroup of a group of real numbers.
advertisement

6. Intersection of subgroups is a ___________
a) group
b) subgroup
c) semigroup
d) cyclic group
View Answer

Answer: b
Explanation: The subgroup property is intersection closed. An arbitrary (nonempty) intersection of subgroups with this property, also attains the similar property.

7. The group of matrices with determinant _________ is a subgroup of the group of invertible matrices under multiplication.
a) 2
b) 3
c) 1
d) 4
View Answer

Answer: c
Explanation: The group of real matrices with determinant 1 is a subgroup of the group of invertible real matrices, both equipped with matrix multiplication. It has to be shown that the product of two matrices with determinant 1 is another matrix with determinant 1, but this is immediate from the multiplicative property of the determinant. This group is usually denoted by(n, R).

8. What is a circle group?
a) a subgroup complex numbers having magnitude 1 of the group of nonzero complex elements
b) a subgroup rational numbers having magnitude 2 of the group of real elements
c) a subgroup irrational numbers having magnitude 2 of the group of nonzero complex elements
d) a subgroup complex numbers having magnitude 1 of the group of whole numbers
View Answer

Answer: a
Explanation: The set of complex numbers with magnitude 1 is a subgroup of the nonzero complex numbers associated with multiplication. It is called the circle group as its elements form the unit circle.

9. A normal subgroup is ____________
a) a subgroup under multiplication by the elements of the group
b) an invariant under closure by the elements of that group
c) a monoid with same number of elements of the original group
d) an invariant equipped with conjugation by the elements of original group
View Answer

Answer: d
Explanation: A normal subgroup is a subgroup that is invariant under conjugation by any element of the original group that is, K is normal if and only if gKg-1=K for any g belongs to G Equivalently, a subgroup K of G is normal if and only if gK=Kg for any g belongs to G.Normal subgroups are useful in constructing quotient groups and in analyzing homomorphisms.

10. Two groups are isomorphic if and only if __________ is existed between them.
a) homomorphism
b) endomorphism
c) isomorphism
d) association
View Answer

Answer: c
Explanation: Two groups M and K are isomorphic (M ~= K) if and only if there exists an isomorphism between them. An isomorphism f:M -> K between two groups M and K is a mapping which satisfies two conditions: 1) f is a bijection and 2) for every x,y belongs to M, we have f(x*My) = f(x) * Kf(y).
advertisement

Sanfoundry Global Education & Learning Series – Discrete Mathematics.

To practice all areas of Discrete Mathematics, here is complete set of 1000+ Multiple Choice Questions and Answers.

advertisement
Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn