This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Groups – Cosets”.
1. a * H is a set of _____ coset.
a) right
b) left
c) sub
d) semi
View Answer
Explanation: Let (H, *) be the semigroup of the group (G, *). Let a belongs to G. (a * H) is the set of a left coset of H in G and (H * a) be the set of a right coset of H in G.
2. a * H = H * a relation holds if __________
a) H is semigroup of an abelian group
b) H is monoid of a group
c) H is a cyclic group
d) H is subgroup of an abelian group
View Answer
Explanation: If h is the subgroup of an abelian group G, then the set of left cosets of H in G is to be set of right cosets i.e, a * H = H * a. Hence, subgroup is called the normal subgroup.
3. Lagrange’s theorem specifies __________
a) the order of semigroup is finite
b) the order of the subgroup divides the order of the finite group
c) the order of an abelian group is infinite
d) the order of the semigroup is added to the order of the group
View Answer
Explanation: Lagrange’s theorem satisfies that the order of the subgroup divides the order of the finite group.
4. A function is defined by f(x)=2x and f(x + y) = f(x) + f(y) is called _____________
a) isomorphic
b) homomorphic
c) cyclic group
d) heteromorphic
View Answer
Explanation: Let (G,*) and (G’,+) are two groups. The mapping f:G->G’ is said to be isomorphism if two conditions are satisfied 1) f is one-to-one function and onto function and 2) f satisfies homomorphism.
5. An isomorphism of a group onto itself is called ____________
a) homomorphism
b) heteromorphism
c) epimorphism
d) automorphism
View Answer
Explanation: An automorphism is defined as an isomorphism of a group onto itself. Similarly, the homomorphism of a group onto itself is defined as the endomorphism of the group.
6. The elements of a vector space form a/an ____________ under vector addition.
a) abelian group
b) commutative group
c) associative group
d) semigroup
View Answer
Explanation: An example of a coset is associated with the theory of vector spaces. The elements (vectors) form an abelian group under the vector addition in a vector space. Subspaces of a vector space are subgroups of this group.
7. A set of representatives of all the cosets is called _________
a) transitive
b) reversal
c) equivalent
d) transversal
View Answer
Explanation: A coset representative is a representative in the equivalence class. In all cosets, a set of the representative is always transversal.
8. Which of the following statement is true?
a) The set of all rational negative numbers forms a group under multiplication
b) The set of all matrices forms a group under multiplication
c) The set of all non-singular matrices forms a group under multiplication
d) The set of matrices forms a subgroup under multiplication
View Answer
Explanation: Since multiplication of two negative rational numbers gives a positive number. Hence, closure property is not satisfied. Singular matrices do not form a group under multiplication. Matrices have to be non-singular (determinant !=0) for the inverse to exist. Hence the set of all non-singular matrices forms a group under multiplication is a true option.
9. How many different non-isomorphic Abelian groups of order 8 are there?
a) 5
b) 4
c) 2
d) 3
View Answer
Explanation: The number of Abelian groups of order Pm (let, P is prime) is the number of partitions of m. Here order is 8 i.e. 23 and so partition of 3 are {1, 1} and {3, 0}. So number of different abelian groups are 2.
10. Consider the set B* of all strings over the alphabet set B = {0, 1} with the concatenation operator for strings ________
a) does not form a group
b) does not have the right identity element
c) forms a non-commutative group
d) forms a group if the empty string is removed from
View Answer
Explanation: Identity element for concatenation is an empty string. Now, we cannot concatenate any string with a given string to get empty string there is no inverse for string concatenation. Only other 3 group properties such as closure, associative and existence of identity are satisfied.
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