Q. If a * b = a such that a ∗ (b ∗ c) = a ∗ b = a and (a * b) * c = a * b = a then ________

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Q. The set of odd and even positive integers closed under multiplication is ________

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Q. If F is a free semigroup on a set S, then the concatenation of two even words is ________

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Q. The set of rational numbers form an abelian group under _________

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Q. Condition of semigroup homomorphism should be ____________

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Q. A function f:(M,∗)→(N,×) is a homomorphism if ______

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Q. A function defined by f(x)=2*x such that f(x+y)=2x+y under the group of real numbers, then ________

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Q. If x * y = x + y + xy then (G, *) is _____________

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Q. Let (A7, ⊗7)=({1, 2, 3, 4, 5, 6}, ⊗7) is a group. It has two sub groups X and Y. X={1, 3, 6}, Y={2, 3, 5}. What is the order of union of subgroups?

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Q. A relation (34 × 78) × 57 = 57 × (78 × 34) can have __________ property.

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Q. B₁: ({0, 1, 2….(n-1)}, xₘ) where xₘ stands for “multiplication-modulo-n” and B₂: ({0, 1, 2….n}, xₙ) where xₙ stands for “multiplication-modulo-m” are the two statements. Both B₁ and B₂ are considered to be __________

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Q. If group G has 65 elements and it has two subgroups namely K and L with order 14 and 30. What can be order of K intersection L?

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Q. Consider the binary operations on X, a*b = a+b+4, for a, b ∈ X. It satisfies the properties of _______

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Q. Let * be the binary operation on the rational number given by a*b=a+b+ab. Which of the following property does not exist for the group?

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Q. Let G be a finite group with two sub groups M & N such that |M|=56 and |N|=123. Determine the value of |M⋂N|.

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Q. A group G, ({0}, +) under addition operation satisfies which of the following properties?

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Q. If (M, *) is a cyclic group of order 73, then number of generator of G is equal to ______

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Q. The set of even natural numbers, {6, 8, 10, 12,..,} is closed under addition operation. Which of the following properties will it satisfy?

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Q. In a group there must be only __________ element.

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Q. _____ is the multiplicative identity of natural numbers.

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