This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Groups – Subgroups”.
1. A trivial subgroup consists of ___________
a) Identity element
c) Inverse element
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.
2. Minimum subgroup of a group is called _____________
a) a commutative subgroup
b) a lattice
c) a trivial group
d) a monoid
Explanation: The subgroups of any given group form a complete lattice under inclusion termed as a 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 __________
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.
b) Existence of inverse for every element
c) Existence of Identity
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
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.
6. Intersection of subgroups is a ___________
d) cyclic group
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.
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
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
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.
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).
Sanfoundry Global Education & Learning Series – Discrete Mathematics.
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