This set covers algebraic structures as taught in Discrete Mathematics: semigroups, monoids, groups, cyclic groups and generators, Lagrange's theorem and cosets, permutation groups, rings and fields such as Z_n, divisor lattices and Boolean algebra. The questions focus on element orders, counting generators, complements in lattices and checking which axioms hold. These are the facts tested in GATE CS, UGC NET and university papers, and they come up again in cryptography courses.
How to practise: read each question and tap the option you think is correct. You will see right away whether you got it, along with a short explanation, and your score is tracked at the bottom of the screen. This set has 18 questions.
Topics covered in this set
- Semigroups, monoids and groups
- Cyclic groups, generators and element order
- Subgroups, cosets and Lagrange's theorem
- Permutation groups
- Rings, integral domains and fields (Z_n)
- Lattices, divisor lattices and complements
- Boolean algebra laws
Discrete Mathematics MCQs with answers and explanations
Q1. What is the order of the element 8 in the group (Z_12, +)?
a) 4
b) 8
c) 3
d) 12
Q2. A group G has order 20. Which of these CANNOT be the order of a subgroup of G?
a) 4
b) 5
c) 10
d) 8
Q3. How many generators does a cyclic group of order 12 have?
a) 4
b) 6
c) 3
d) 11
Q4. Which statement is true for every group of prime order p?
a) It is non-abelian
b) It has exactly p subgroups
c) It is cyclic
d) It has a proper non-trivial subgroup
Q5. Which element generates the multiplicative group Z_7* = {1, 2, 3, 4, 5, 6}?
a) 2
b) 6
c) 4
d) 3
Q6. In a group G, every element x satisfies x^2 = e. What must G be?
a) Abelian
b) Cyclic
c) Of odd order
d) Non-abelian
Q7. In any group, what is (ab)^(-1) equal to?
a) a^(-1) b^(-1)
b) b^(-1) a^(-1)
c) ab
d) ba
Q8. For an integer n ≥ 2, the ring (Z_n, +, ×) is a field if and only if:
a) n is odd
b) n is even
c) n is a perfect square
d) n is prime
Q9. How many units (invertible elements) does the ring Z_15 have?
a) 8
b) 7
c) 14
d) 4
Q10. How many non-zero zero divisors does the ring Z_12 have?
a) 4
b) 7
c) 8
d) 5
Q11. For which n is the lattice of positive divisors of n, ordered by divisibility, a Boolean algebra?
a) 12
b) 36
c) 30
d) 20
Q12. In the lattice D_30 (divisors of 30 under divisibility), what is the complement of 5?
a) 6
b) 1
c) 30
d) 10
Q13. In the lattice of positive divisors of 36 under divisibility, what are 4 ∨ 6 (join) and 4 ∧ 6 (meet)?
a) 24 and 2
b) 12 and 2
c) 12 and 1
d) 36 and 2
Q14. In a Boolean algebra, what does (x + y)(x + y') simplify to?
a) x
b) y
c) x + y
d) 1
Q15. The number of elements in a finite Boolean algebra is always:
a) a prime number
b) a power of 2
c) a multiple of 3
d) a perfect square
Q16. What is the order of the permutation (1 2 3)(4 5) in S5?
a) 5
b) 3
c) 6
d) 2
Q17. H is a subgroup of order 4 in a group G of order 24. How many distinct left cosets of H are there in G?
a) 4
b) 20
c) 96
d) 6
Q18. What kind of algebraic structure is (N ∪ {0}, +), the non-negative integers under addition?
a) A group
b) A monoid but not a group
c) A semigroup with no identity
d) An abelian group
Quick revision notes
- In (Z_n, +), the order of a is n / gcd(a, n).
- A cyclic group of order n has φ(n) generators and one subgroup for each divisor of n.
- Lagrange: the order of a subgroup divides |G|; the number of cosets is |G|/|H|.
- Every group of prime order is cyclic and hence abelian.
- (ab)^(-1) = b^(-1) a^(-1); if x^2 = e for all x, the group is abelian.
- Z_n is a field iff n is prime; units of Z_n number φ(n).
- The order of a permutation is the lcm of its disjoint cycle lengths.
- D_n is a Boolean algebra iff n is square-free; a finite Boolean algebra has 2^k elements.
Frequently asked questions
What is the difference between a semigroup, a monoid and a group?
A semigroup is a set with an associative binary operation. A monoid also has an identity element. A group additionally requires every element to have an inverse.
Does the converse of Lagrange's theorem hold?
Not in general. A divisor of |G| need not be the order of a subgroup: the alternating group A4 has order 12 but no subgroup of order 6. The converse does hold for cyclic groups, and Sylow's theorems give it for prime-power divisors.
When is a lattice a Boolean algebra?
A lattice is a Boolean algebra when it is bounded, distributive and complemented. In a distributive lattice, complements are unique when they exist.
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