Discrete Mathematics Questions and Answers – Group Theory

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This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Group Theory”.

1. A non empty set A is termed as an algebraic structure ________
a) with respect to binary operation *
b) with respect to ternary operation ?
c) with respect to binary operation +
d) with respect to unary operation –
View Answer

Answer: a
Explanation: A non empty set A is called an algebraic structure w.r.t binary operation “*” if (a*b) belongs to S for all (a*b) belongs to S. Therefore “*” is closure operation on ‘A’.
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2. An algebraic structure _________ is called a semigroup.
a) (P, *)
b) (Q, +, *)
c) (P, +)
d) (+, *)
View Answer

Answer: a
Explanation: An algebraic structure (P,*) is called a semigroup if a*(b*c) = (a*b)*c for all a,b,c belongs to S or the elements follow associative property under “*”. (Matrix,*) and (Set of integers,+) are examples of semigroup.

3. Condition for monoid is __________
a) (a+e)=a
b) (a*e)=(a+e)
c) a=(a*(a+e)
d) (a*e)=(e*a)=a
View Answer

Answer: d
Explanation: A Semigroup (S,*) is defined as a monoid if there exists an element e in S such that (a*e) = (e*a) = a for all a in S. This element is called identity element of S w.r.t *.

4. A monoid is called a group if _______
a) (a*a)=a=(a+c)
b) (a*c)=(a+c)
c) (a+c)=a
d) (a*c)=(c*a)=e
View Answer

Answer: d
Explanation: A monoid(B,*) is called Group if to each element there exists an element c such that (a*c)=(c*a)=e. Here e is called an identity element and c is defined as the inverse of the corresponding element.

5. A group (M,*) is said to be abelian if ___________
a) (x+y)=(y+x)
b) (x*y)=(y*x)
c) (x+y)=x
d) (y*x)=(x+y)
View Answer

Answer: b
Explanation: A group (M,*) is said to be abelian if (x*y) = (x*y) for all x, y belongs to M. Thus Commutative property should hold in a group.
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6. Matrix multiplication is a/an _________ property.
a) Commutative
b) Associative
c) Additive
d) Disjunctive
View Answer

Answer: b
Explanation: The set of two M*M non-singular matrices form a group under matrix multiplication operation. Since matrix multiplication is itself associative, it holds associative property.

7. A cyclic group can be generated by a/an ________ element.
a) singular
b) non-singular
c) inverse
d) multiplicative
View Answer

Answer: a
Explanation: A singular element can generate a cyclic group. Every element of a cyclic group is a power of some specific element which is known as a generator ‘g’.

8. How many properties can be held by a group?
a) 2
b) 3
c) 5
d) 4
View Answer

Answer: c
Explanation: A group holds five properties simultaneously –
i) Closure
ii) associative
iii) Commutative
iv) Identity element
v) Inverse element.

9. A cyclic group is always _________
a) abelian group
b) monoid
c) semigroup
d) subgroup
View Answer

Answer: a
Explanation: A cyclic group is always an abelian group but every abelian group is not a cyclic group. For instance, the rational numbers under addition is an abelian group but is not a cyclic one.
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10. {1, i, -i, -1} is __________
a) semigroup
b) subgroup
c) cyclic group
d) abelian group
View Answer

Answer: c
Explanation: The set of complex numbers {1, i, -i, -1} under multiplication operation is a cyclic group. Two generators i and -i will covers all the elements of this group. Hence, it is a cyclic group.

Sanfoundry Global Education & Learning Series – Discrete Mathematics.

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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