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This quiz contains multiple-choice problems on group theory and axioms, closure and associativity, subgroups, identity and inverse existence, Burnside’s theorem, cyclic and permutation groups.
Let (A7, ⊗7) = ({1, 2, 3, 4, 5, 6}, ⊗7) is a group. It has two subgroups, X and Y, where X={1, 3, 6}, Y={2, 3, 5}. What is the order of the union of subgroups?
65
5
32
18
B1: ({0, 1, 2….(n-1)}, xm), where xn stands for “multiplication-modulo-n”
B2: ({0, 1, 2….n}, xn), where xn stands for “multiplication-modulo-m”
Both B1 and B2 are considered to be
Groups
Semigroups
Subgroups
Associative subgroups
If group G has 65 elements and two subgroups, namely K and L, with orders 14 and 30, respectively, what can be the order of K intersection L?
10
42
5
35
Consider the binary operations on X for a, b ∈ X, a*b = a + b + 4. It satisfies the properties of
Abelian group
Semigroup
Multiplicative group
Isomorphic group
Let * be the binary operation on the rational number given by a*b = a + b + ab. Which of the following properties does not exist for the group?
Closure property
Identity property
Symmetric property
Associative property
Let G be a finite group with two subgroups, M and N, such that |M| = 56 and |N| = 123. Determine the value of |M⋂N|.
1
56
14
78
A group G, ({0}, +) under addition operation satisfies which of the following properties?
Identity, multiplicity and inverse
Closure, associativity, inverse and identity
Multiplicity, associativity and closure
Inverse and closure
If (M, *) is a cyclic group of order 73, then the number of generators of G is equal to
89
23
72
17
The set of even natural numbers, {6, 8, 10, 12,..,} is closed under addition operation. Which of the following properties will it satisfy?
Closure property
Associative property
Symmetric property
Identity property
A trivial subgroup consists of
Identity element
Coset
Inverse element
Ring
The minimum subgroup of a group is called a
Commutative subgroup
Lattice
Trivial group
Monoid
Let K be a group with eight 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
8
2
3
4
__ is not necessarily a group property
Commutativity
Existence of inverse for every element
Existence of Identity
Associativity
A group of rational numbers is an example of a
Subgroup of a group of integers
Subgroup of a group of real numbers
Subgroup of a group of irrational numbers
Subgroup of a group of complex numbers
A relation (34 × 78) × 57 = 57 × (78 × 34) can have the __ property.
Distributive
Associative
Commutative
Closure
